Question

Determine whether the geometric series is convergent or divergent. if it is convergent, find the sum. (if the quantity diverges, enter diverges. ) 7 − 8 64 7 − 512 49

Answer 1

The sum of the given series is 49 and it converges to infinity.

According to the statement

we have given that a series which is 7, -8, 64/7, 512/49

And we have to find that the series is converges or diverges.

So, For this purpose,

The nth term in the series is 6 multiplied by the (n-1)th power of -8/7:

So,

$a_{1} = 7(\frac{-8}{7} )^{1-1}$

$a_{2} = -8(\frac{-8}{7} )^{2-1}$

$a_{3} = \frac{64}{7} (\frac{-8}{7} )^{3-1}$

And so on then

Sum of the series become to the nth partial sum

$S_{N} = 7(1 +\frac{-8}{7} + ......+ (\frac{-8^(n-2)}{7^(n-2)}) + (\frac{-8^(n-1)}{7^(n-1)}))$

Multiplying both sides by -8/7 gives

$\frac{-8}{7} S_{N} = 7(\frac{-8}{7} +\frac{-8^{2} }{7^{2}} + ......+ (\frac{-8^(n-1)}{7^(n-1)}) + (\frac{-8^n}{7^n}))$

and subtracting this from $S_{N}$ gives

$\frac{-1}{7} S_{N} = 7(1-\frac{-8^n}{7^n} )$

$S_{N} = 49 (-1 + (8/7)^n)$

Now the sum of the series is 49 and the it converges to infinity.

So, The sum of the given series is 49 and it converges to infinity.

Learn more about the Convergence of series here

https://brainly.com/question/23804543

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