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Wize University Calculus 1 Textbook > Integrals

Finite Sums

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Finite Sums

When adding up a large number of terms, it's often helpful to use Summation (Sigma) Notation.

Finite Sums

The symbol
∑i=1nxi\boxed{\qquad\sum_{i=1}^{n}x_i\qquad}
denotes the finite sum of all numbers xix_i, where ii varies from 1 to nn.

∑i=15i=1+2+3+4+5\displaystyle \sum_{i=1}^5i=1+2+3+4+5

Important Finite Sums

(1) ∑i=1ni=n(n+1)2(2) ∑i=1ni2=n(n+1)(2n+1)6(3) ∑i=1ni3=(n(n+1)2)2\text{(1)}\, \boxed{\sum_{i=1}^{n}i=\frac{n(n+1)}{2}}\quad \text{(2)}\,\boxed{\sum_{i=1}^{n}i^2=\frac{n(n+1)(2n+1)}{6}}\quad \text{(3)}\,\boxed{\sum_{i=1}^{n}i^3=\left(\frac{n(n+1)}{2}\right)^2}


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Example: Finite Sums

Evaluate the following sums

a) ∑j=14jsin⁡(jπ2)\displaystyle \sum_{j=1}^{4}j\sin{\left(j\frac{\pi}{2}\right)}

∑j=14jsin⁡(jπ2)=sin⁡(π2)+2sin⁡(π)+3sin⁡(3π2)+4sin⁡(2π)=1+0−3+0=−2\begin{aligned} \sum_{j=1}^4j\sin\left(j\frac{\pi}{2}\right) &= \sin\left(\frac{\pi}{2}\right) + 2\sin\left(\pi\right) + 3\sin\left(3\frac{\pi}{2}\right) + 4\sin\left(2\pi\right)\\ &= 1 +0-3+0\\ &= -2 \end{aligned}



b) ∑j=16(3−j2)\displaystyle \sum_{j=1}^{6}(3-j^{2})

∑j=16(3−j2)=∑j=163−∑j=16j2=3(6)−6(6+1)(2(6)+1)6=18−(7)(13)=−73\begin{aligned} \sum_{j=1}^6(3-j^2) &= \sum_{j=1}^63 - \sum_{j=1}^6j^2\\ &= 3(6) - \dfrac{\cancel{6}(6+1)(2(6)+1)}{\cancel{6}}\\ &= 18 - (7)(13)\\ &= -73 \end{aligned}
Evaluate

∑i=110 [2(i−3i2)−1]\displaystyle \sum_{i=1}^{10}\ \left[2\left(i-3i^2\right)-1\right]
Evaluate ∑i=1020 (3i−4)\displaystyle \sum_{i=10}^{20}\ \left(3i-4\right)


Extra Practice
Finite Sums
Evaluate ∑i=510[2i−3]\displaystyle\sum_{i=5}^{10}\left[2i-3\right].
Finite Sums: Sigma Notation
∑i=030sin⁡(iπ2)\sum_{i=0}^{30}\sin\left(\frac{i\pi}{2}\right) equals
Practice: Telescoping Sum (~F2017 Final Q22)
Practice: Sigma Notation
Evaluate Σi=12000 (1i+1−1i+2)\Sigma_{i=1}^{2000}\ \left(\frac{1}{i+1}-\frac{1}{i+2}\right)
Practice: Sigma Notation (~F2017 Final Q21)
Practice: Sigma Notation
Evaluate Σi=1020 (3i−4)\Sigma_{i=10}^{20}\ \left(3i-4\right).
Finite Sum with Trig
Evaluate the following sum: ∑i=25sin⁡(iπ2)\displaystyle \sum_{i=2}^{5}\sin{\left(i\frac{\pi}{2}\right)}
Finite Sum
Evaluate the following sum: ∑i=1065k(k−1)\displaystyle \sum_{i=10}^{65}k(k-1)
Finite Sum with Trig
Evaluate the sum: ∑j=120jsin⁡(jπ2)\displaystyle \sum_{j=1}^{20}j\sin{\left(j\frac{\pi}{2}\right)}
Finite Sums
Evaluate ∑i=2100 1i−1−1i\sum_{i=2}^{100}\ \frac{1}{i-1}-\frac{1}{i}.
Finite Sums
Evaluate ∑i=510[2i−3]\displaystyle\sum_{i=5}^{10}\left[2i-3\right].
Evaluate 3⋅Σi=510 [(2i)2−2(i−1)]3\cdot\Sigma_{i=5}^{10}\ \left[\left(2i\right)^2-2\left(i-1\right)\right]
Evaluate Σi=1100 (13i−1−13i)\Sigma_{i=1}^{100}\ \left(\frac{1}{3^{i-1}}-\frac{1}{3^i}\right).
Finite Sums
Evaluate
∑i=120(1+4i)\sum_{i=1}^{20}(1+4i)
The sum of the series ∑n=1N(3+∑i=1ni)=\displaystyle \sum_{n=1}^{N} {\left(3+\sum_{i=1}^n{i}\right)}=
Practice: Telescoping Sums
Evaluate ∑i=12000 (1i+1−1i+2)\displaystyle \sum_{i=1}^{2000}\ \left(\frac{1}{i+1}-\frac{1}{i+2}\right)
Practice: Sigma w/ Limits
Evaluate lim⁡n→∞ ∑i=1n(3n4)⋅[i(2i2−1)+1]\displaystyle \lim_{n\rightarrow\infty}\ \sum_{i=1}^n\left(\frac{3}{n^4}\right)\cdot\left[i\left(2i^2-1\right)+1\right]
Practice: Sigma Notation
Evaluate ∑i=1020 (3i−4)\displaystyle \sum_{i=10}^{20}\ \left(3i-4\right).
Finite Sum with Trig
Evaluate the following sum: ∑i=25sin⁡(iπ2)\displaystyle \sum_{i=2}^{5}\sin{\left(i\frac{\pi}{2}\right)}
Finite Sum
Evaluate the following sum: ∑i=1065k(k−1)\displaystyle \sum_{i=10}^{65}k(k-1)
Finite Sum with Trig
Evaluate the sum: ∑j=120jsin⁡(jπ2)\displaystyle \sum_{j=1}^{20}j\sin{\left(j\frac{\pi}{2}\right)}
Evaluate Σi=1100 (13i−1−13i)\Sigma_{i=1}^{100}\ \left(\frac{1}{3^{i-1}}-\frac{1}{3^i}\right).
Evaluate 3⋅Σi=510 [(2i)2−2(i−1)]3\cdot\Sigma_{i=5}^{10}\ \left[\left(2i\right)^2-2\left(i-1\right)\right]
Finite Sums
Evaluate Σi=310 [i(2+i)−1]\Sigma_{i=3}^{10}\ \left[i\left(2+i\right)-1\right]
Finite Sums
Evaluate Σi=11000 1ei−1ei−1\Sigma_{i=1}^{1000}\ \frac{1}{e^i}-\frac{1}{e^{i-1}}.
Finite Sums: Sigma Notation
∑i=030sin⁡(iπ2)\sum_{i=0}^{30}\sin\left(\frac{i\pi}{2}\right) equals
Practice: Sigma Notation (~F2017 Final Q21)
Practice: Sigma Notation
Evaluate Σi=1020 (3i−4)\Sigma_{i=10}^{20}\ \left(3i-4\right).
Practice: Telescoping Sum (~F2017 Final Q22)
Practice: Sigma Notation
Evaluate Σi=12000 (1i+1−1i+2)\Sigma_{i=1}^{2000}\ \left(\frac{1}{i+1}-\frac{1}{i+2}\right)