Question

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

Answer 1

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

Answer 2

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

Answer 3

As a start, write the fraction in single exponent

Answer 4

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

Answer 5

next recall geometric series

Answer 6

a/(1 - r) ??

Answer 7

Yep

Answer 8

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

Answer 9

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

Answer 10

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

Answer 11

and get that value?

Answer 12

there is no equation

Answer 13

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

Answer 14

that's what i mean haha :)

Answer 15

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

Answer 16

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

Answer 17

what do you get for first term, a ?

Answer 18

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

Answer 19

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

Answer 20

-0.375

Answer 21

that's still not right tho haha

Answer 22

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

Answer 23

got it! thanks :)

Answer 24

np