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Does the series converge or diverge? If it converges what is the sum? Show your work

\(\sum_{n=1}^{inf}-4(-1/2)^{n-1}\)

 Jun 6, 2019
 #1
avatar+6180 
+1

\(\sum \limits_{n=0}^\infty \alpha^n = \dfrac{1}{1-\alpha},~\forall \alpha \ni |\alpha|<1\\ \text{and does not converge otherwise}\)

 

\(\sum \limits_{n=1}^\infty ~-4\left(-\dfrac 1 2\right)^{n-1} = -4\left(\sum \limits_{n=0}^\infty~\left(-\dfrac 1 2\right)^n \right)\)

 

I leave it to you to apply the first line to the second.

 Jun 6, 2019
edited by Rom  Jun 6, 2019
 #2
avatar+109560 
+1

List the first few terms

 

-4, 2, -1, 1/2, -1/4, 1/8........

 

The common ratio  is  -1/2    and the first term is -4

 

So....the sum converges to

 

   -4                    -4

_______  =    _____  =   -4 (2/3)  =  -8/3

1 - (-1/2)           (3/2)

 

 

 

 

 

cool cool cool

 Jun 6, 2019

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