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

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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}

Related  If Σ an and Σ bn are series of positive terms and lim an/bn=0 from n to +∞ show by example that Σ an may converge and Σ bn not converge solved problems in calculus

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


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