Question

1/4 - 3/2 + 9 - 54 +.... find the sum of the series if it converges

Answer 1

the common ratio for the series is r = -6

a series converges when |r|<1

Answer 2

how did you figure that out?

Answer 3

the common ratio is found by

\[\frac{T _{2}}{T _{1}} = \frac{T _{3}}{T _{2}} = \frac{T _{4}}{T _{3}} ......\]

Answer 4

if the ratio is the same for the first few terms... then it will be assumed to be the same for all

Answer 5

what is T1, T2, T3, and T4 then?

Answer 6

I mean value wise

Answer 7

T_1 = 1/4
T_2=- 3/2
...

Answer 8

\[T _{1} = \frac{1}{4}, T _{2} = \frac{-3}{2}\]

Answer 9

The sum of the series or a GP is
\[S_n = \frac{a(r^n - 1)}{r-1}\]
a-->1st term
r--->common ration

Substitute the values and you will get
\[S_n = \frac{\frac{1}{4}({(-6)}^n - 1)}{-6-1}\]
\[S_n = \frac{\frac{1}{4}({(-6)}^n - 1)}{-7}\]
\[S_n = \frac{{((-6)}^n - 1)}{-28}\]

Answer 10

*of

Answer 11

Now you can find out whether it converges or not

Answer 12

the common ratio for the series is r = -6

a series converges when |r|<1

Answer 13

As Campbell said. So no need of doing all this. Just determine the r and find out. The above I did was just for explaining

Answer 14

thanks