A calculus problem by Snehal Shekatkar

Calculus Level 3

Consider the following series:

∑ n = 1 ∞ ( n − 1 n ) n = 0 + ( 1 2 ) 2 + ( 2 3 ) 3 + ⋯ \sum_{n=1}^{\infty}\left(\frac{n-1}{n}\right)^{n} = 0 + \left(\frac{1}{2}\right)^{2}+ \left(\frac{2}{3}\right)^{3} + \cdots

Which of the following statements is true for the series above?

The series converges to a value greater than 10000 The series converges to 10000 The series diverges to infinity The series converges to a value less than 10000

Snehal Shekatkar by Snehal Shekatkar , 32, India

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1 solution

Snehal Shekatkar
Jul 6, 2016

Since the convergence or divergence of any series is decided by the large-n terms, let us analyze the behavior of s n = ( n − 1 n ) n s_{n}=\left(\frac{n-1}{n}\right)^{n} as n → ∞ n\to\infty . We have:

lim ⁡ n → ∞ s n = lim ⁡ n → ∞ = ( n − 1 n ) n = ( 1 − 1 n ) n = 1 e \lim_{n\to\infty}s_{n}=\lim_{n\to\infty} = \left(\frac{n-1}{n}\right)^{n} = \left(1-\frac{1}{n}\right)^{n} = \frac{1}{e}

Where e e is Euler's number Thus, each of the large-n terms contributes a finite amount to the sum of the series and hence the series diverges.

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