# Show that the series converges. What is the value of the series? sum_{n=2}^infty(-frac{5}{3})^n(frac{2}{5})^{n+1}

Show that the series converges. What is the value of the series?
$$\sum_{n=2}^\infty(-\frac{5}{3})^n(\frac{2}{5})^{n+1}$$

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We have to Show that the series converges. What is the value of the series?
Series is given below:
$$\Rightarrow\sum_{n=2}^\infty(-\frac{5}{3})^n(\frac{2}{5})^{n+1}$$
We will use geometric series test .
Geometriic series test says if series of the form $$ar^{n-1}$$ and $$|r|<1$$ then we can say that series is converges and sum of series is given by S.
Work is shown below:
$$\Rightarrow\sum_{n=1}^\infty ar^{n-1}$$
$$|r|<1$$
$$\Rightarrow S_{\infty}=\frac{a}{1-r}$$
Now compare the given series to geometric series and find value of r .
If $$r<1$$ then series converges.
In this case $$r=\frac23 <1$$ so series converge.
After that find value of a (starting term of series).
With the help of a, r and sum of geometric series formula we will find sum.
Work is shown below:
$$\Rightarrow\sum_{n=2}^\infty(-\frac53)^n(\frac25)^{n+1}$$
$$\Rightarrow\sum_{n=2}^\infty(-\frac53)^n(\frac25)^{n+1}$$
$$\Rightarrow\sum_{n=2}^\infty\frac25(-1)^n(\frac23)^n$$
$$\Rightarrow|r|=|-\frac23|$$
$$|r|=\frac23<1$$
Previous step continue
$$\Rightarrow_{n=2}^\infty\frac25(-1)^n(\frac23)^n$$
$$\Rightarrow\frac{8}{45}-\frac{16}{135}+\frac{32}{405}...$$
$$\Rightarrow a=\frac{8}{45}$$
$$\Rightarrow r=-\frac{2}{3}$$
$$\Rightarrow S_{\infty}=\frac{\frac{8}{45}}{1+\frac23}$$
$$=\frac{8}{75}$$
$$\Rightarrow S_{\infty}=\frac{8}{75}$$

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