Answer to Question #168225 in Calculus for Festus

Question

Given a series 1 + 1/2 + 1/4 + 1/8 + 1/16+......; determine the nature of the series, the sum of the n term of the series and sim of the terms as n tends to infinity.

Answer 1

An infinite geometric series is the sum of numbers in geometric progression. A geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers in which each term after the first is found by multiplying the previous one by a fixed non-zero number (common ratio):

$$\sum_{k=0}^\infin a_k =\sum_{k=0}^\infin a \cdot r^k=[if\space |r|<1]=\frac a {1-r}$$

The sum of the first n terms of a geometric progression is determined by the expression:

$$\sum_{k=0}^n a \cdot r^k=a\frac {1-r^n} {1-r}, where \space r\ne1$$

If the common ratio is r=1/2 and the scale factor is a=1 then

$$\sum_{k=0}^\infin a \cdot r^k=\sum_{k=0}^\infin \frac 1 {2^k}=1+\frac 1 2+\frac 1 4+\frac 1 8+\frac 1 {16}+...$$

and the sum of the n term of the series is

$$\sum_{k=0}^n \frac 1 {2^k}=\frac {1-(\frac 1 2)^n} {1-\frac 1 2}=2-\frac 1 {2^{n-1}}$$

and sum of the terms as n tends to infinity is

$$\sum_{k=0}^\infin \frac 1 {2^k}=\frac 1 {1-\frac 1 2}=2$$