Find the values of x for which the series x + x^3  + x^5  + • • • converges, and express the sum as a function of x.  infinite series problems and solutions | solved problems in calculus

Solution

The geometric sum formula for infinite terms is given as:

If  common ration |r| < 1, S_\infty=\frac{a}{1-r}

If |r| > 1, the series does not converge and it has no sum.

Where

a is the first term of series

r is the common ratio of series

n is the number of terms of series

This is a geometric series with ratio x^2 Hence, it converges for |x^2|<1  that is, for |x|<1 By the formula a/(I – r) for the sum of a geometric series, the sum is \frac{x}{1-x^2}

infinite series problems with solutions