Question

For the series ∑n=0 → ∞ (4ˣ -7)ⁿ, find the interval of convergence and what the series converges to.

Answer 1

This series is geometric; substitute
`y=4^x-7` to see how.

Then the series converges if

`|4^x-7|<1\implies-1<4^x-7<1\implies6<4^x<8\implies\log_46<x<\log_48`

Under these conditions, we have

`\displaystyle\sum_(n=0)^\infty(4^x-7)^n=\frac1{1-(4^x-7)}=\frac1{8-4^x}`