Answered

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To test this series for convergence

∝
∑ (4^n+7)/7^n
n=1
You could use the limit comparison test, comparing it to the series
∝
∑ r^n
n=1

Where r=???
Completing the test, it shows the series:

Sagot :

LammettHash

r should be 4/7. You're comparing the given series to a geometric series that converges. By the limit comparison test, you have

[tex]\displaystyle\lim_{n\to\infty}\frac{\frac{4^n+7}{7^n}}{\frac{4^n}{7^n}}=\lim_{n\to\infty}\frac{4^n+7}{4^n}=\lim_{n\to\infty}\left(1+\frac7{4^n}\right)=1[/tex]

and since this limit is positive and finite, and the series you're comparing to is convergent, then the first series must also be convergent.

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