Answer to Question #213799 in Calculus for Aroosha ch

Question

A geometric progression has first term and common ratio

i)                  Find the set of values of x for which geometric progression has a sum to infinity.

• ii)                Find the value of x for which the sum to infinity of geometric progression 

Answer 1

I haven't got any information on first term and common ratio of the geometric progression. I assume that the first term is $a\ne 0$ and common ratio is x.

Then n-th term is $ax^{n-1}$ and sum of the first n term is

$s_n=a_1+\dots+a_n=a+ax+\dots+ax^{n-1}=a\frac{x^n-1}{x-1}=\frac{a}{1-x}-\frac{a}{1-x}x^n$ ($x\ne 1$)

$s_n$ tends to a finite limit as $n\to+\infty$ if and only if $x^n$ tends to a finite limit as $n\to+\infty$, i.e., if and only if $|x|<1$. With this condition we have $\lim\limits_{n\to+\infty}x^n=0$ and $\sum\limits_{n=1}^{+\infty}ax^{n-1}=\lim\limits_{n\to+\infty}s_n=\frac{a}{1-x}$.

In the case x=1 we have $a_n=ax^{n-1}=a$ and $s_n=an$ diverges to infinity.

Answer. (i) $|x|<1$; (ii) $\sum\limits_{n=1}^{+\infty}ax^{n-1}=\frac{a}{1-x}$.