## Math2400 one year ago determine the sum of this series...i thought it was divergent but I guess it diverges cuz it says that it's not divergent

1. Math2400

$\sum_{n=1}^{\infty} \frac{ (-3)^{n-1} }{ 5^n}$

2. Math2400

@jim_thompson5910 would u be able to help me on this please?

3. ganeshie8

As a start, write the fraction in single exponent

4. ganeshie8

$\sum_{n=1}^{\infty} \frac{ (-3)^{n-1} }{ 5^n} = \sum_{n=1}^{\infty} \frac{ (-3)^{n} }{ (-3)5^n} = -\frac{1}{3}\sum_{n=1}^{\infty} \left(\frac{ -3 }{ 5}\right)^n$

5. ganeshie8

next recall geometric series

6. Math2400

a/(1 - r) ??

7. ganeshie8

Yep

8. Math2400

what would be r? this is where I'm confused

9. ganeshie8

r = the stuff under the exponent = -3/5

10. Math2400

and for a i just plug in 1 into the equation?

11. Math2400

and get that value?

12. ganeshie8

there is no equation

13. ganeshie8

for "a", just plugin n=1 into the general term of series

14. Math2400

that's what i mean haha :)

15. ganeshie8

i kno, just making sure you see the difference between "equation" and "expression"

16. Math2400

okay i plugged everything in but it's zero??

17. ganeshie8

what do you get for first term, a ?

18. Math2400

(-3)^(1-1)/5^1=0

19. ganeshie8

Ah no, below is our slightly massaged series : $-\frac{1}{3}\color{blue}{\sum_{n=1}^{\infty} \left(\frac{ -3 }{ 5}\right)^n}$ plugin $$n=1$$, you get $$a_1 = -\frac{3}{5}$$ $$r = -\frac{3}{5}$$ so the infinite sum is $\dfrac{-3/5}{1-(-3/5)}$ simplify

20. Math2400

-0.375

21. Math2400

that's still not right tho haha

22. ganeshie8

looks good, don't forget that -1/3 in front

23. Math2400

got it! thanks :)

24. ganeshie8

np