Basic Analysis I — Real Analysis · Problem 2.6.10

Sequences and Series — Jiri Lebl

Mathematics — Real Analysis · power-series-radius-of-convergence

Problem

a) Prove that lim⁡n→∞n1/n=1\lim_{n \to \infty} n^{1/n} = 1 using the following procedure: Write n1/n=1+bnn^{1/n} = 1 + b_n and note bn>0b_n > 0. Then show that (1+bn)n≥n(n−1)2bn2(1 + b_n)^n \geq \frac{n(n-1)}{2} b_n^2 and use this to show that lim⁡n→∞bn=0\lim_{n \to \infty} b_n = 0. b) Use the result of part a) to show that if ∑n=0∞anxn\sum_{n=0}^{\infty} a_n x^n is a convergent power series with radius of convergence RR, then ∑n=0∞nanxn\sum_{n=0}^{\infty} n a_n x^n is also convergent with the same radius of convergence.

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