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Wize AP Calculus (BC) Textbook > Integration & Accumulation of Change

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)