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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.

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    Hint: do the terms go to $0$? – 2017-02-16
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    Individually, wouldn't each term would go to infinity? But because there are more exponents in the denominator, it grows faster than the numerator and thus causes the whole thing to go to 0. Is this the correct line of reasoning? – 2017-02-16
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    Not really,try dividing denominator and numerator by $5^i$ or by $7^i$ – 2017-02-16
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    It's always a good idea to simply write out terms...a computer can quickly give you the first dozen, or the first fifty for that matter. At least then you'll know what it is you are trying to prove. – 2017-02-16

3 Answers 3

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HINT, by the ratio test:

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

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    Ah, the ratio test, I always forget about that one. I got too caught up on trying to play with the exponents! Thank you very much! – 2017-02-16
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    @JonathonM Haha, just don't forget him :) you're welcome, I'm glad that I could help you! – 2017-02-16
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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.

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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.