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) and is defined as:
i=a∑bf(i)=f(a)+f(a+1)+f(a+2)+⋯+f(b)
Lower limit (i=a): Starting point of the sum.
Upper limit (i=b): Ending point of the sum.
General term (f(i)): Expression dependent on the index i.
Special case:
If a>b, the summation is considered empty, and its result is 0.
Basic Summation
i=1∑nai=a1+a2+⋯+an
Examples
∑k=14k=1+2+3+4=10
∑j=13(2j)=2+4+6=12
∑i=15i2=1+4+9+16+25=55
∑m=13m1=1+21+31≈1.833
∑n=023n=1+3+9=13
∑p=14(−1)p=−1+1−1+1=0.
Properties of Summations
Important
Linearity:
i=1∑n(c⋅ai±d⋅bi)=ci=1∑nai±di=1∑nbi
Examples
∑k=13(2k+3)=2∑k=13k+∑k=133=2(6)+9=21
∑j=12(5j−j2)=5∑j=12j−∑j=12j2=5(3)−(5)=10
∑i=14(3i2+2i)=3∑i=14i2+2∑i=14i=3(30)+2(10)=110
∑m=13(2m+4)=21∑m=13m+∑m=134=26+12=15
∑n=15(4−n)=∑n=154−∑n=15n=20−15=5
∑p=12(3p3−2p)=3∑p=12p3−2∑p=12p=3(9)−2(3)=21.
Summation of a Constant:
i=1∑nc=n⋅c
Examples
∑k=157=5×7=35
∑j=13(−2)=3×(−2)=−6
∑i=14π=4π
∑m=160=0
∑n=122=22
∑p=11021=10×21=5.
Range Decomposition:
i=1∑nai=i=1∑kai+i=k+1∑nai(1≤k<n)
Notable Summations
Important
Sum of the first n natural numbers:
k=1∑nk=2n(n+1)
Examples
∑k=110k=210×11=55
∑k=1100k=5050
∑k=115k=215×16=120
∑k=11k=1 (base case)
∑k=10k=0 (lower limit > upper limit)
∑k=58k=5+6+7+8=26 (no direct formula, calculated manually).
Sum of squares:
k=1∑nk2=6n(n+1)(2n+1)
Examples
∑k=13k2=63×4×7=14
∑k=15k2=65×6×11=55
∑k=110k2=385
∑k=11k2=1
∑k=10k2=0
∑k=24k2=4+9+16=29 (no direct formula for arbitrary limits).
m=2,n=3,r=2: ∑k=02(k2)(2−k3)=3+6+3=12=(25) (intentional error for verification).
m=5,n=5,r=5: ∑k=05(k5)(5−k5)=(510)=252.
Infinite Summations (Series)
Harmonic series:
k=1∑∞k1(Divergent)
Telescoping series:
k=1∑n(f(k+1)−f(k))=f(n+1)−f(1)
Conditional Summations
1≤i≤ni even∑ai=k=1∑⌊n/2⌋a2k
Summations and Products
i=1∑nln(ai)=ln(i=1∏nai)
Summation by Parts (General Formula)
k=1∑nakΔbk=an+1bn+1−a1b1−k=1∑nbk+1Δak
where Δak=ak+1−ak.
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) and defined as:
i=a∏bf(i)=f(a)×f(a+1)×f(a+2)×⋯×f(b)
Lower limit (i=a): Starting point of the product.
Upper limit (i=b): Ending point of the product.
General term (f(i)): Expression dependent on the index i.
Special case:
If a>b, the product is considered empty, and its result is 1 (by mathematical convention).
Basic Product
i=1∏nai=a1×a2×⋯×an
Examples
∏k=13k=1×2×3=6
∏j=14(j+1)=2×3×4×5=120
∏i=12(2i)=2×4=8
∏m=151=1×1×1×1×1=1
∏n=13(−1)n=(−1)×1×(−1)=1
∏p=24p2=4×9×16=576.
Properties of Products
Important
Linearity (with constant exponents):
i=1∏n(aic⋅bid)=(i=1∏nai)c⋅(i=1∏nbi)d
Product of a Constant:
i=1∏nc=cn
Examples
∏k=142=24=16
∏j=13(−1)=(−1)3=−1
∏i=15π=π5
∏m=123=(3)2=3
∏n=105=1 (empty product convention)
∏p=1621=(21)6=641.
Range Decomposition:
i=1∏nai=(i=1∏kai)⋅(i=k+1∏nai)(1≤k<n)
Multiplicative Variable Substitution:
i=a∏bf(i)=j=a±k∏b±kf(j∓k)
Notable Products
Important
Factorial:
n!=k=1∏nk
Examples
4!=1×2×3×4=24
1!=1
0!=1 (standard definition)
5!/3!=6120=20
3!×2!=6×2=12
2!⋅4!6!=2×24720=15.
Double Factorial:
n!!=k=0∏⌈n/2⌉−1(n−2k)
Product of Terms in Arithmetic Progression:
k=0∏n−1(a+k⋅d)=dn⋅Γ(a/d)Γ(a/d+n)
where Γ is the gamma function.
Product of Terms in Geometric Progression:
k=0∏n−1a⋅rk=an⋅r2n(n−1)
Examples
∏k=032⋅3k=24⋅36=16×729=11,664
∏j=025⋅(21)j=53⋅(21)3=125×81=15.625
∏i=041⋅2i=15⋅210=1,024
∏m=0110⋅(−1)m=102⋅(−1)1=−100
∏n=03π⋅en=π4⋅e6
∏p=05c⋅rp=c6⋅r15.
Fundamental Relation
Both operations are related via the natural logarithm (ln):
ln(i=a∏bf(i))=i=a∑bln(f(i))
This allows converting products into summations for simplification.