A Weird Square Root

Calculus Level 3

Let $\displaystyle f(x)= 1 + \frac{x}{2}+\frac{x^2}{4}+\frac{x^3}{8}+\cdots$ , for $-1\le x \le 1$ .

Find $\large \sqrt{e^{\int_0^1 f(x)\, dx}}$ .

0

\ln 2

2

1

1 solution

Hana Wehbi
Nov 8, 2016

We know that $f(x)$ is an infinite geometric series, thus the infinite sum is $\frac{1}{1-\frac{x}{2}} = \frac{2}{2-x}$ .

Thus, $\int_{0}^{1}\frac{2}{2-x} = 2\ln 2$ .

Then $\sqrt{e^{2\ln2}}= \sqrt{2^2}= 2$ .