Understanding the Convergence of Infinite Series

[Jim Alamia]

Hello everyone,

I’m currently studying infinite series in my calculus course and have encountered some concepts that I find quite challenging, particularly the criteria for convergence. I’ve read about the Comparison Test, Ratio Test, and Root Test, but I’m struggling to apply these tests effectively in different scenarios.

To give you some context, I’ve worked through a few examples, but I often find myself unsure about which test to use or how to justify my choice. For instance, when faced with a series like ∑n=1∞n22n\sum_{n=1}^{\infty} \frac{n^2}{2^n}∑n=1∞2nn2, I initially thought the Ratio Test would be appropriate, but I’m not confident in my analysis.

Could someone provide guidance on the following:

How do you determine which convergence test to apply to a given series?

Are there any tips or tricks for recognizing when a series converges or diverges?

Can you provide examples of series where different tests yield different insights?

I believe this information will not only help me but also others who might be grappling with similar issues. Thank you for your assistance!

[joigus]

You only get to develop some intuition after you've done a bunch of examples.

Do you mean,
\[ \sum_{n=1}^{\infty}\frac{n^{2}}{2^{n}} \]?

If that's the series you're referring to, the quotient is a good way to go.

You generally use comparison when your general term is easily related to a well-known convergent of divergent series, like the geometric series, the harmonic, etc. The root criterion I would try when I have a function of n raised to a function of n. But it's not an easy subject in which you can give a fixed recipe.

Code:

\[ \sum_{n=1}^{\infty}\frac{n^{2}}{2^{n}} \]

[joigus]

Was this of any help at all?