Question

evaluate the geometric series or state that it diverges
sum_{k=3}^{infty} (4^k)/7^k

Answer 1

\[\sum_{k=3}^{\infty} (4^k)/7^k \]

Answer 2

you can write it as
to get the geometric sequence,
since the argument inside the brackets is lower than 1, we know that the series converges
but since its geometric we can also calculate the value of the sum.

sum of a geo series is
so you plig the 4/7 there
you can 1 over 3/7 which is 7 over 3
but you count from k=3 and the geo formula counts from k = 0 if im not mistaken

so you need to subtract the k=0, k=1 and k=2 terms

to get those you can simply do k=0 is (4/7) ^0 and so on