AP Calculus AB and BC glossary

Index of summation

Also called: Summation index, Dummy index

The index of summation is the counter variable in sigma notation together with its starting value below the sigma. It is a placeholder, so renaming it changes nothing, and shifting where it starts is allowed provided the terms are rewritten to compensate, leaving the sum unchanged.

The index is a dummy variable, so ∑n=1∞1n2\sum_{n=1}^{\infty} \frac{1}{n^2} and ∑k=1∞1k2\sum_{k=1}^{\infty} \frac{1}{k^2} are the same number. One useful consequence: the summation variable can never survive into the value of the sum, so if you summed over ii, no ii can remain in the answer. An upper limit is a different thing. ∑i=1ni=n(n+1)2\sum_{i=1}^{n} i = \frac{n(n+1)}{2} keeps its nn because nn is the stopping value, not the index.

To move the start from n=1n = 1 down to n=0n = 0, replace every nn inside with n+1n+1. That turns ∑n=1∞xnn\sum_{n=1}^{\infty} \frac{x^n}{n} into ∑n=0∞xn+1n+1\sum_{n=0}^{\infty} \frac{x^{n+1}}{n+1}, the same series rewritten to line up with a standard form.

Differentiating a power series term by term kills the constant term, so ddx∑n=0∞cnxn=∑n=1∞ncnxn−1\frac{d}{dx} \sum_{n=0}^{\infty} c_n x^n = \sum_{n=1}^{\infty} n c_n x^{n-1}. The starting index moves up by one because the n=0n = 0 term differentiates to zero.

The mistake

Sliding the starting value without rewriting the terms. Going from ∑n=1∞xn\sum_{n=1}^{\infty} x^n to ∑n=0∞xn\sum_{n=0}^{\infty} x^n quietly adds 11 to the sum. Each step down in the lower limit must be paid for with an n+1n+1 inside.

Appears in: Unit 6: Integration and Accumulation, Unit 10: Infinite Sequences and Series (BC)