Tea Time Numerical Analysis · Problem 1.2.32

Preliminaries — Leon Brin

Mathematics — Numerical Analysis · taylor-series

Problem

The Taylor series for

f(x) = e^x

T(x) = 1 + x + \frac{1}{2!}x^2 + \frac{1}{3!}x^3 + \frac{1}{4!}x^4 + \frac{1}{5!}x^5 + \cdots

This series converges to

f(x)

for all values of

x

. In particular, for

x = 1

, this means that

f(1) = T(1) = 1 + 1 + \frac{1}{2!}(1)^2 + \frac{1}{3!}(1)^3 + \frac{1}{4!}(1)^4 + \cdots

Simplifying this equation, we see that

e = 1 + 1 + \frac{1}{2} + \frac{1}{6} + \frac{1}{24} + \frac{1}{120} + \cdots

Use Taylor Series to find infinite sums that sum to ((a))

\ln(2)

((b))

2/3

((c))

\pi/4

((d))

\sqrt{2}