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Find the sum of the convergent series. (round your answer to four decimal places. ) [infinity] (sin(9))n n = 1

Sagot :

The sum of the convergent series [tex]\sum_{n=1}^{\infty}~(sin(1))^n[/tex] is 5.31

For given question,

We have been given a series [tex]\sum_{n=1}^{\infty}~(sin(1))^n[/tex]

[tex]\sum_{n=1}^{\infty}~(sin(1))^n=sin(1)+(sin(1))^2+...+(sin(1))^n[/tex]

We need to find the sum of given convergent series.

Given series is a geometric series with ratio r = sin(1)

The first term of the given geometric series is [tex]a_1=sin(1)[/tex]

So, the sum is,

= [tex]\frac{a_1}{1-r}[/tex]

= sin(1) / [1 - sin(1)]

This means, the series converges to sin(1) / [1 - sin(1)]

[tex]\sum_{n=1}^{\infty}~(sin(1))^n[/tex]

= [tex]\frac{sin(1)}{1-sin(1)}[/tex]

= [tex]\frac{0.8415}{1-0.8415}[/tex]

= [tex]\frac{0.8415}{0.1585}[/tex]

= 5.31

Therefore, the sum of the convergent series [tex]\sum_{n=1}^{\infty}~(sin(1))^n[/tex] is 5.31

Learn more about the convergent series here:

https://brainly.com/question/15415793

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