Mathematical Methods in the Physical Sciences · Problem 4.1

INFINITE SERIES, POWER SERIES — Mary Boas

Mathematics — Methods · convergent-divergent-series

Problem

A careful mathematical definition of a convergent infinite series with sum

S

is this: Given any small positive number (usually called

\epsilon

by mathematicians), it is possible to find an integer

N

so that

|S - S_n| < \epsilon

for every

n \geq N

. In words this means that no matter how tiny an

\epsilon

you think of, all the partial sums

S_n

n

terms differ less than your

\epsilon

from the final sum

S

if you add up

N

terms or more. Select some

\epsilon

's and find the corresponding

N

's for the following series:

\displaystyle\sum_{n=1}^{\infty} \frac{1}{2^n}