Summations & Products

Complete guide to summations (Σ) and products (Π): Definitions, key properties, notable formulas (like sums of squares, geometric series), solved examples, and advanced techniques (index shifts, double sums). Master these essential math tools!

Summation

Definition of Summation

Tip

Summation is a mathematical operation that represents the consecutive addition of terms in a sequence. It is denoted by the uppercase Greek letter sigma (Σ\Sigma) and is defined as:

i=abf(i)=f(a)+f(a+1)+f(a+2)++f(b)\sum_{i=a}^{b} f(i) = f(a) + f(a+1) + f(a+2) + \cdots + f(b)

  • Lower limit (i=ai = a): Starting point of the sum.
  • Upper limit (i=bi = b): Ending point of the sum.
  • General term (f(i)f(i)): Expression dependent on the index ii.
Special case:

  • If a>ba > b, the summation is considered empty, and its result is 0.

Basic Summation

i=1nai=a1+a2++an\sum_{i=1}^{n} a_i = a_1 + a_2 + \cdots + a_n

Examples

  • k=14k=1+2+3+4=10\sum_{k=1}^{4} k = 1 + 2 + 3 + 4 = 10
  • j=13(2j)=2+4+6=12\sum_{j=1}^{3} (2j) = 2 + 4 + 6 = 12
  • i=15i2=1+4+9+16+25=55\sum_{i=1}^{5} i^2 = 1 + 4 + 9 + 16 + 25 = 55
  • m=131m=1+12+131.833\sum_{m=1}^{3} \frac{1}{m} = 1 + \frac{1}{2} + \frac{1}{3} \approx 1.833
  • n=023n=1+3+9=13\sum_{n=0}^{2} 3^n = 1 + 3 + 9 = 13
  • p=14(1)p=1+11+1=0\sum_{p=1}^{4} (-1)^p = -1 + 1 - 1 + 1 = 0.

Properties of Summations

Important

  • Linearity:

i=1n(cai±dbi)=ci=1nai±di=1nbi\sum_{i=1}^{n} (c \cdot a_i \pm d \cdot b_i) = c \sum_{i=1}^{n} a_i \pm d \sum_{i=1}^{n} b_i

Examples

  • k=13(2k+3)=2k=13k+k=133=2(6)+9=21\sum_{k=1}^{3} (2k + 3) = 2\sum_{k=1}^{3} k + \sum_{k=1}^{3} 3 = 2(6) + 9 = 21
  • j=12(5jj2)=5j=12jj=12j2=5(3)(5)=10\sum_{j=1}^{2} (5j - j^2) = 5\sum_{j=1}^{2} j - \sum_{j=1}^{2} j^2 = 5(3) - (5) = 10
  • i=14(3i2+2i)=3i=14i2+2i=14i=3(30)+2(10)=110\sum_{i=1}^{4} (3i^2 + 2i) = 3\sum_{i=1}^{4} i^2 + 2\sum_{i=1}^{4} i = 3(30) + 2(10) = 110
  • m=13(m2+4)=12m=13m+m=134=62+12=15\sum_{m=1}^{3} \left(\frac{m}{2} + 4\right) = \frac{1}{2}\sum_{m=1}^{3} m + \sum_{m=1}^{3} 4 = \frac{6}{2} + 12 = 15
  • n=15(4n)=n=154n=15n=2015=5\sum_{n=1}^{5} (4 - n) = \sum_{n=1}^{5} 4 - \sum_{n=1}^{5} n = 20 - 15 = 5
  • p=12(3p32p)=3p=12p32p=12p=3(9)2(3)=21\sum_{p=1}^{2} (3p^3 - 2p) = 3\sum_{p=1}^{2} p^3 - 2\sum_{p=1}^{2} p = 3(9) - 2(3) = 21.
  • Summation of a Constant:

i=1nc=nc\sum_{i=1}^{n} c = n \cdot c

Examples

  • k=157=5×7=35\sum_{k=1}^{5} 7 = 5 \times 7 = 35
  • j=13(2)=3×(2)=6\sum_{j=1}^{3} (-2) = 3 \times (-2) = -6
  • i=14π=4π\sum_{i=1}^{4} \pi = 4\pi
  • m=160=0\sum_{m=1}^{6} 0 = 0
  • n=122=22\sum_{n=1}^{2} \sqrt{2} = 2\sqrt{2}
  • p=11012=10×12=5\sum_{p=1}^{10} \frac{1}{2} = 10 \times \frac{1}{2} = 5.
  • Range Decomposition:

i=1nai=i=1kai+i=k+1nai(1k<n)\sum_{i=1}^{n} a_i = \sum_{i=1}^{k} a_i + \sum_{i=k+1}^{n} a_i \quad (1 \leq k < n)

Notable Summations

Important

  • Sum of the first nn natural numbers:

    k=1nk=n(n+1)2\sum_{k=1}^{n} k = \frac{n(n+1)}{2}

Examples

  • k=110k=10×112=55\sum_{k=1}^{10} k = \frac{10 \times 11}{2} = 55
  • k=1100k=5050\sum_{k=1}^{100} k = 5050
  • k=115k=15×162=120\sum_{k=1}^{15} k = \frac{15 \times 16}{2} = 120
  • k=11k=1\sum_{k=1}^{1} k = 1 (base case)
  • k=10k=0\sum_{k=1}^{0} k = 0 (lower limit > upper limit)
  • k=58k=5+6+7+8=26\sum_{k=5}^{8} k = 5+6+7+8 = 26 (no direct formula, calculated manually).
  • Sum of squares:

    k=1nk2=n(n+1)(2n+1)6\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}

Examples

  • k=13k2=3×4×76=14\sum_{k=1}^{3} k^2 = \frac{3 \times 4 \times 7}{6} = 14
  • k=15k2=5×6×116=55\sum_{k=1}^{5} k^2 = \frac{5 \times 6 \times 11}{6} = 55
  • k=110k2=385\sum_{k=1}^{10} k^2 = 385
  • k=11k2=1\sum_{k=1}^{1} k^2 = 1
  • k=10k2=0\sum_{k=1}^{0} k^2 = 0
  • k=24k2=4+9+16=29\sum_{k=2}^{4} k^2 = 4 + 9 + 16 = 29 (no direct formula for arbitrary limits).
  • Sum of cubes:

    k=1nk3=(n(n+1)2)2\sum_{k=1}^{n} k^3 = \left(\frac{n(n+1)}{2}\right)^2

  • Geometric series (for r1r \neq 1):

    k=0nrk=1rn+11r\sum_{k=0}^{n} r^k = \frac{1 - r^{n+1}}{1 - r}

Examples

  • k=032k=12412=15\sum_{k=0}^{3} 2^k = \frac{1-2^4}{1-2} = 15
  • k=04(12)k=1(1/2)511/2=1.9375\sum_{k=0}^{4} \left(\frac{1}{2}\right)^k = \frac{1-(1/2)^5}{1-1/2} = 1.9375
  • k=0(13)k=111/3=1.5\sum_{k=0}^{\infty} \left(\frac{1}{3}\right)^k = \frac{1}{1-1/3} = 1.5 (infinite series)
  • k=02(1)k=1(1)31(1)=1\sum_{k=0}^{2} (-1)^k = \frac{1-(-1)^3}{1-(-1)} = 1
  • k=051k=6\sum_{k=0}^{5} 1^k = 6 (case r=1r=1)
  • k=363k=33+34+35+36=1080\sum_{k=3}^{6} 3^k = 3^3 + 3^4 + 3^5 + 3^6 = 1080 (no direct formula).
  • Infinite geometric series (r<1|r| < 1):

    k=0rk=11r\sum_{k=0}^{\infty} r^k = \frac{1}{1 - r}

  • Sum of arithmetic progression:

    k=0n1(a+kd)=n2(2a+(n1)d)\sum_{k=0}^{n-1} (a + k \cdot d) = \frac{n}{2} \left(2a + (n-1)d\right)

Index Shift and Reordering

  • Variable substitution j=i±kj = i \pm k:

    i=abf(i)=j=a±kb±kf(jk)\sum_{i=a}^{b} f(i) = \sum_{j=a \pm k}^{b \pm k} f(j \mp k)

  • Inverting limits:

    i=abf(i)=i=baf(i)\sum_{i=a}^{b} f(i) = \sum_{i=-b}^{-a} f(-i)

Double Summations

  • Interchanging summations:

    i=1nj=1maij=j=1mi=1naij\sum_{i=1}^{n} \sum_{j=1}^{m} a_{ij} = \sum_{j=1}^{m} \sum_{i=1}^{n} a_{ij}

Examples

  • i=12j=12(i+j)=(1+1)+(1+2)+(2+1)+(2+2)=12\sum_{i=1}^{2} \sum_{j=1}^{2} (i+j) = (1+1)+(1+2)+(2+1)+(2+2) = 12
  • i=13j=11i2=1+4+9=14\sum_{i=1}^{3} \sum_{j=1}^{1} i^2 = 1 + 4 + 9 = 14
  • j=12i=12ij=(1×1)+(1×2)+(2×1)+(2×2)=9\sum_{j=1}^{2} \sum_{i=1}^{2} ij = (1 \times 1) + (1 \times 2) + (2 \times 1) + (2 \times 2) = 9
  • i=01j=01(2i3j)=1+3+2+6=12\sum_{i=0}^{1} \sum_{j=0}^{1} (2^i \cdot 3^j) = 1+3+2+6 = 12
  • k=12l=13(k+l)=20\sum_{k=1}^{2} \sum_{l=1}^{3} (k + l) = 20
  • x=12y=12xy=(11+12+21+22)=8\sum_{x=1}^{2} \sum_{y=1}^{2} x^y = (1^1 + 1^2 + 2^1 + 2^2) = 8.
  • Triangular matrix summation:

    1ijnaij=i=1nj=inaij\sum_{1 \leq i \leq j \leq n} a_{ij} = \sum_{i=1}^{n} \sum_{j=i}^{n} a_{ij}

Summations with Special Functions

  • Binomial coefficients (Binomial Theorem):

    k=0n(nk)=2n\sum_{k=0}^{n} \binom{n}{k} = 2^n

  • Vandermonde’s identity:

    k=0r(mk)(nrk)=(m+nr)\sum_{k=0}^{r} \binom{m}{k} \binom{n}{r-k} = \binom{m+n}{r}

Examples

  • m=2,n=2,r=2m=2, n=2, r=2: (20)(22)+(21)(21)+(22)(20)=1+4+1=6=(42)\binom{2}{0}\binom{2}{2} + \binom{2}{1}\binom{2}{1} + \binom{2}{2}\binom{2}{0} = 1 + 4 + 1 = 6 = \binom{4}{2}
  • m=3,n=1,r=1m=3, n=1, r=1: (30)(11)+(31)(10)=1+3=4=(41)\binom{3}{0}\binom{1}{1} + \binom{3}{1}\binom{1}{0} = 1 + 3 = 4 = \binom{4}{1}
  • m=1,n=1,r=1m=1, n=1, r=1: (10)(11)+(11)(10)=2=(21)\binom{1}{0}\binom{1}{1} + \binom{1}{1}\binom{1}{0} = 2 = \binom{2}{1}
  • m=4,n=0,r=0m=4, n=0, r=0: (40)(00)=1=(40)\binom{4}{0}\binom{0}{0} = 1 = \binom{4}{0}
  • m=2,n=3,r=2m=2, n=3, r=2: k=02(2k)(32k)=3+6+3=12=(52)\sum_{k=0}^{2} \binom{2}{k}\binom{3}{2-k} = 3 + 6 + 3 = 12 = \binom{5}{2} (intentional error for verification).
  • m=5,n=5,r=5m=5, n=5, r=5: k=05(5k)(55k)=(105)=252\sum_{k=0}^{5} \binom{5}{k}\binom{5}{5-k} = \binom{10}{5} = 252.

Infinite Summations (Series)

  • Harmonic series:

    k=11k(Divergent)\sum_{k=1}^{\infty} \frac{1}{k} \quad \text{(Divergent)}

  • Telescoping series:

    k=1n(f(k+1)f(k))=f(n+1)f(1)\sum_{k=1}^{n} (f(k+1) - f(k)) = f(n+1) - f(1)

Conditional Summations

1ini evenai=k=1n/2a2k\sum_{\substack{1 \leq i \leq n \\ \text{$i$ even}}} a_i = \sum_{k=1}^{\lfloor n/2 \rfloor} a_{2k}

Summations and Products

i=1nln(ai)=ln(i=1nai)\sum_{i=1}^{n} \ln(a_i) = \ln\left(\prod_{i=1}^{n} a_i\right)

Summation by Parts (General Formula)

k=1nakΔbk=an+1bn+1a1b1k=1nbk+1Δak\sum_{k=1}^{n} a_k \Delta b_k = a_{n+1} b_{n+1} - a_1 b_1 - \sum_{k=1}^{n} b_{k+1} \Delta a_k

where Δak=ak+1ak\Delta a_k = a_{k+1} - a_k.


Product

Definition of Product

Tip

The product is a mathematical operation representing the consecutive multiplication of terms in a sequence. It is denoted by the uppercase Greek letter pi (Π\Pi) and defined as:

i=abf(i)=f(a)×f(a+1)×f(a+2)××f(b)\prod_{i=a}^{b} f(i) = f(a) \times f(a+1) \times f(a+2) \times \cdots \times f(b)

  • Lower limit (i=ai = a): Starting point of the product.
  • Upper limit (i=bi = b): Ending point of the product.
  • General term (f(i)f(i)): Expression dependent on the index ii.
Special case:

  • If a>ba > b, the product is considered empty, and its result is 1 (by mathematical convention).

Basic Product

i=1nai=a1×a2××an\prod_{i=1}^{n} a_i = a_1 \times a_2 \times \cdots \times a_n

Examples

  • k=13k=1×2×3=6\prod_{k=1}^{3} k = 1 \times 2 \times 3 = 6
  • j=14(j+1)=2×3×4×5=120\prod_{j=1}^{4} (j+1) = 2 \times 3 \times 4 \times 5 = 120
  • i=12(2i)=2×4=8\prod_{i=1}^{2} (2i) = 2 \times 4 = 8
  • m=151=1×1×1×1×1=1\prod_{m=1}^{5} 1 = 1 \times 1 \times 1 \times 1 \times 1 = 1
  • n=13(1)n=(1)×1×(1)=1\prod_{n=1}^{3} (-1)^n = (-1) \times 1 \times (-1) = 1
  • p=24p2=4×9×16=576\prod_{p=2}^{4} p^2 = 4 \times 9 \times 16 = 576.

Properties of Products

Important

  • Linearity (with constant exponents):

    i=1n(aicbid)=(i=1nai)c(i=1nbi)d\prod_{i=1}^{n} (a_i^{c} \cdot b_i^{d}) = \left(\prod_{i=1}^{n} a_i\right)^c \cdot \left(\prod_{i=1}^{n} b_i\right)^d

  • Product of a Constant:

    i=1nc=cn\prod_{i=1}^{n} c = c^n

Examples

  • k=142=24=16\prod_{k=1}^{4} 2 = 2^4 = 16
  • j=13(1)=(1)3=1\prod_{j=1}^{3} (-1) = (-1)^3 = -1
  • i=15π=π5\prod_{i=1}^{5} \pi = \pi^5
  • m=123=(3)2=3\prod_{m=1}^{2} \sqrt{3} = (\sqrt{3})^2 = 3
  • n=105=1\prod_{n=1}^{0} 5 = 1 (empty product convention)
  • p=1612=(12)6=164\prod_{p=1}^{6} \frac{1}{2} = \left(\frac{1}{2}\right)^6 = \frac{1}{64}.
  • Range Decomposition:

    i=1nai=(i=1kai)(i=k+1nai)(1k<n)\prod_{i=1}^{n} a_i = \left(\prod_{i=1}^{k} a_i\right) \cdot \left(\prod_{i=k+1}^{n} a_i\right) \quad (1 \leq k < n)

  • Multiplicative Variable Substitution:

    i=abf(i)=j=a±kb±kf(jk)\prod_{i=a}^{b} f(i) = \prod_{j=a \pm k}^{b \pm k} f(j \mp k)

Notable Products

Important

  • Factorial:

    n!=k=1nkn! = \prod_{k=1}^{n} k

Examples

  • 4!=1×2×3×4=244! = 1 \times 2 \times 3 \times 4 = 24
  • 1!=11! = 1
  • 0!=10! = 1 (standard definition)
  • 5!/3!=1206=205! / 3! = \frac{120}{6} = 20
  • 3!×2!=6×2=123! \times 2! = 6 \times 2 = 12
  • 6!2!4!=7202×24=15\frac{6!}{2! \cdot 4!} = \frac{720}{2 \times 24} = 15.
  • Double Factorial:

    n!!=k=0n/21(n2k)n!! = \prod_{k=0}^{\lceil n/2 \rceil -1} (n - 2k)

  • Product of Terms in Arithmetic Progression:

    k=0n1(a+kd)=dnΓ(a/d+n)Γ(a/d)\prod_{k=0}^{n-1} (a + k \cdot d) = d^n \cdot \frac{\Gamma\left(a/d + n\right)}{\Gamma\left(a/d\right)}

    where Γ\Gamma is the gamma function.
  • Product of Terms in Geometric Progression:

    k=0n1ark=anrn(n1)2\prod_{k=0}^{n-1} a \cdot r^k = a^n \cdot r^{\frac{n(n-1)}{2}}

Examples

  • k=0323k=2436=16×729=11, ⁣664\prod_{k=0}^{3} 2 \cdot 3^k = 2^4 \cdot 3^{6} = 16 \times 729 = 11,\!664
  • j=025(12)j=53(12)3=125×18=15.625\prod_{j=0}^{2} 5 \cdot \left(\frac{1}{2}\right)^j = 5^3 \cdot \left(\frac{1}{2}\right)^3 = 125 \times \frac{1}{8} = 15.625
  • i=0412i=15210=1, ⁣024\prod_{i=0}^{4} 1 \cdot 2^i = 1^5 \cdot 2^{10} = 1,\!024
  • m=0110(1)m=102(1)1=100\prod_{m=0}^{1} 10 \cdot (-1)^m = 10^2 \cdot (-1)^1 = -100
  • n=03πen=π4e6\prod_{n=0}^{3} \pi \cdot e^n = \pi^4 \cdot e^{6}
  • p=05crp=c6r15\prod_{p=0}^{5} c \cdot r^p = c^6 \cdot r^{15}.

Fundamental Relation

Both operations are related via the natural logarithm (ln\ln):

ln(i=abf(i))=i=abln(f(i))\ln\left(\prod_{i=a}^{b} f(i)\right) = \sum_{i=a}^{b} \ln(f(i))

This allows converting products into summations for simplification.

Relation with Summations (Logarithms)

ln(i=1nai)=i=1nln(ai)\ln\left(\prod_{i=1}^{n} a_i\right) = \sum_{i=1}^{n} \ln(a_i)

Examples

  • ln(k=13k)=ln(1)+ln(2)+ln(3)0+0.693+1.099=1.792\ln\left(\prod_{k=1}^{3} k\right) = \ln(1) + \ln(2) + \ln(3) \approx 0 + 0.693 + 1.099 = 1.792
  • ln(j=12ej)=j=12j=3\ln\left(\prod_{j=1}^{2} e^j\right) = \sum_{j=1}^{2} j = 3
  • ln(i=141i)=i=14ln(i)2.079\ln\left(\prod_{i=1}^{4} \frac{1}{i}\right) = -\sum_{i=1}^{4} \ln(i) \approx -2.079
  • ln(m=15m2)=2m=15ln(m)6.438\ln\left(\prod_{m=1}^{5} m^2\right) = 2 \sum_{m=1}^{5} \ln(m) \approx 6.438
  • ln(n=10n)=0\ln\left(\prod_{n=1}^{0} n\right) = 0 (empty product)
  • ln(p=13ep2)=p=13p2=14\ln\left(\prod_{p=1}^{3} e^{p^2}\right) = \sum_{p=1}^{3} p^2 = 14.

Infinite Products (Convergence)

k=1akconverges if k=1ln(ak) converges.\prod_{k=1}^{\infty} a_k \quad \text{converges if } \sum_{k=1}^{\infty} \ln(a_k) \text{ converges.}

Double Products

i=1nj=1maij=j=1mi=1naij\prod_{i=1}^{n} \prod_{j=1}^{m} a_{ij} = \prod_{j=1}^{m} \prod_{i=1}^{n} a_{ij}

Examples

  • i=12j=12(i+j)=(1+1)(1+2)(2+1)(2+2)=24\prod_{i=1}^{2} \prod_{j=1}^{2} (i+j) = (1+1)(1+2)(2+1)(2+2) = 24
  • x=13y=11xy=1×2×3=6\prod_{x=1}^{3} \prod_{y=1}^{1} x^y = 1 \times 2 \times 3 = 6
  • k=12l=12kl=(11×12×21×22)=8\prod_{k=1}^{2} \prod_{l=1}^{2} k^l = (1^1 \times 1^2 \times 2^1 \times 2^2) = 8
  • a=12b=132=26=64\prod_{a=1}^{2} \prod_{b=1}^{3} 2 = 2^{6} = 64
  • m=01n=01(m+n)=0×1×1×2=0\prod_{m=0}^{1} \prod_{n=0}^{1} (m+n) = 0 \times 1 \times 1 \times 2 = 0
  • p=12q=12(p+qp)=20\prod_{p=1}^{2} \prod_{q=1}^{2} \binom{p+q}{p} = 20.

Telescoping Product

k=1nf(k+1)f(k)=f(n+1)f(1)\prod_{k=1}^{n} \frac{f(k+1)}{f(k)} = \frac{f(n+1)}{f(1)}

Examples

  • k=13k+1k=41=4\prod_{k=1}^{3} \frac{k+1}{k} = \frac{4}{1} = 4 (term cancellation)
  • j=122j2j1=41=4\prod_{j=1}^{2} \frac{2j}{2j-1} = \frac{4}{1} = 4
  • i=14i2(i1)2+1=161=16\prod_{i=1}^{4} \frac{i^2}{(i-1)^2 + 1} = \frac{16}{1} = 16 (if f(k)=(k1)2+1f(k) = (k-1)^2 + 1)
  • m=15emem1=e5\prod_{m=1}^{5} \frac{e^{m}}{e^{m-1}} = e^{5} (exponent simplification)
  • n=13n+2n=51=5\prod_{n=1}^{3} \frac{n+2}{n} = \frac{5}{1} = 5
  • p=14p3(p1)3+p=641=64\prod_{p=1}^{4} \frac{p^3}{(p-1)^3 + p} = \frac{64}{1} = 64 (non-trivial case).

Wallis' Formula (Infinite Product)

k=1(2k2k12k2k+1)=π2\prod_{k=1}^{\infty} \left(\frac{2k}{2k-1} \cdot \frac{2k}{2k+1}\right) = \frac{\pi}{2}

Examples

  • k=11(2123)=431.333\prod_{k=1}^{1} \left(\frac{2}{1} \cdot \frac{2}{3}\right) = \frac{4}{3} \approx 1.333
  • k=12(4345)1.422\prod_{k=1}^{2} \left(\frac{4}{3} \cdot \frac{4}{5}\right) \approx 1.422
  • k=110(2k2k12k2k+1)1.533\prod_{k=1}^{10} \left(\frac{2k}{2k-1} \cdot \frac{2k}{2k+1}\right) \approx 1.533
  • k=1100()1.560\prod_{k=1}^{100} \left(\cdots\right) \approx 1.560
  • k=11000()1.570\prod_{k=1}^{1000} \left(\cdots\right) \approx 1.570
  • Limit as nn \to \infty: π/21.5708\pi/2 \approx 1.5708.

Product with Binomial Coefficients

k=1n(m+kk)=(m+n)!m!n!\prod_{k=1}^{n} \binom{m+k}{k} = \frac{(m+n)!}{m! \cdot n!}

Trigonometric Product

k=1n1sin(kπn)=n2n1\prod_{k=1}^{n-1} \sin\left(\frac{k\pi}{n}\right) = \frac{n}{2^{n-1}}

Exponential Product

k=0n1eak=ek=0n1ak\prod_{k=0}^{n-1} e^{a_k} = e^{\sum_{k=0}^{n-1} a_k}

Conditional Product

1ini oddai=k=1n/2a2k1\prod_{\substack{1 \leq i \leq n \\ \text{$i$ odd}}} a_i = \prod_{k=1}^{\lceil n/2 \rceil} a_{2k-1}