Answer :
The interval of convergence of the power series is (-10/7, 10/7).
What does Interval of convergence tell you?
A group of x-values on which a power series converges is known as the interval of convergence. In order to create a convergent series, you can plug in this interval of x-values.
The interval of convergence of a power series is the set of values of x for which the series converges. To find the interval of convergence of the given power series, we can use the ratio test.
The ratio test states that if the series [tex]∑ a_nx^n[/tex] has a positive radius of convergence R, then the series converges for |x| < R and diverges for |x| > R. The radius of convergence is given by the formula
[tex]R = 1/limsup|a_n+1/a_n|.[/tex]
In the given power series, the coefficient [tex]a_n[/tex] is given by [tex]a_n = (7x)^n/(10n)!.[/tex]
We can rewrite this as [tex]a_n = x^n(7/10)^n/n!.[/tex]
Taking the limit of the ratio [tex]a_n+1/a_n[/tex] as n goes to infinity, we get:
[tex]limsup|a_n+1/a_n| \\\\= limsup|x^(n+1)(7/10)^(n+1)/(n+1)!/(x^n(7/10)^n/n!)|\\\\= limsup|x(7/10)/(n+1)|\\\\= |x(7/10)|[/tex]
Therefore, the radius of convergence R is [tex]1/|x(7/10)|[/tex].
This means that the interval of convergence is [tex]|x| < 1/|7x/10| = 10/7.[/tex]
We can check the endpoints of the interval to see if the series converges at these points. At x = 10/7, the series becomes [tex](7(10/7))^n/(10n)! = 1^n/(10n)!.[/tex]
This series is a geometric series with first term 1 and common ratio 1/10, and it converges because [tex]|1/10| < 1.[/tex]
At x = -10/7, the series becomes[tex](7(-10/7))^n/(10n)! = (-1)^n/(10n)![/tex]. This series is an alternating series with first term 1 and common ratio (-1/10), and it also converges because [tex]|-1/10| < 1.[/tex]
Therefore, The interval of convergence of the power series is (-10/7, 10/7).
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