Simplifying expressions containing exponents to multiplebases

Question

The question asks to determine whether the following is convergent or divergent. If convergent, find what it converges to.

$$\sum_{i=1}^\infty \frac{7^i}{3^{i-2}-5^{i+1}}$$

My first step was to get everything into the same power if i. This resulted in:

$$\sum_{i=1}^\infty \frac{7^i}{\frac{1}{9}3^{i}-(5)5^{i}}$$

However from here, I do not know where to go. I am assuming my goal is to get this to look like a geometric sequence $\frac{1}{r^i}$, however I don't know how to proceed.

Answer 1

HINT, by the ratio test:

$$\lim_{\text{n}\to\infty}\left|\frac{\frac{7^{\text{n}+1}}{3^{\left(\text{n}+1\right)-2}-5^{\left(\text{n}+1\right)+1}}}{\frac{7^\text{n}}{3^{\text{n}-2}-5^{\text{n}+1}}}\right|=\frac{7}{3}\cdot\lim_{\text{n}\to\infty}\left|\frac{3^\text{n}-9\cdot5^{1+\text{n}}}{3^\text{n}-3\cdot5^{2+\text{n}}}\right|=\frac{7}{3}\cdot\frac{3}{5}=\frac{7}{5}\color{red}{>}1$$

Answer 2

For sufficiently large $i$ , $7^i> \mid 3^{i-1}-5^{i+1} \mid $ so the modulus terms will become larger and larger and so the series will diverge.

Answer 3

You have managed to get the first step wrong, by subtracting $2$ and $1$ from the exponents instead of adding. A proper equivalent form is $$ 7\sum_{k=0}^\infty \frac{7^k}{27\cdot 3^k - 25\cdot 5^k} $$ Since for large $k$, the term ratio between one term and the one before goes to $\frac75$, this does not converge. That is fortunate, because in general expressions like $$ \sum_{k=0}^\infty \frac{a^k}{m\cdot b^k - n\cdot c^k} $$ for $b \neq c$ and $\max(b,c)>a$ do converge but have no simple closed form.