Calculus: why the infinite series 1/n^2, n ε N, converges
Tutoring calculus, infinite series come up. The tutor mentions one way to realize the convergence of the infinite series 1/n^2.
The series 1/n^2, meaning 1/1 + 1/4 + 1/9 + 1/16 + 1/25 + …. converges; even though it has infinite terms, its sum is a number rather than infinity. In particular, its sum is π^2/6.
The integral test says that a series, evaluating a function over all the natural numbers, will diverge if the definite integral of its function from 1 to infinity also does. Moreover, if its corresponding integral converges, so will it.
∫1∞1/x^2 = -1/x |1∞ = 1, so it definitely converges. This is one reason we could expect the series 1/n^2 to converge even if we didn’t know its sum.
Source:
Larson, Roland E. and Robert P. Hostetler. Calculus. Toronto: D C Heath and Company, 1989.
Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.
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