calculus - Infinite series help

Are there any hard and fast rules for finding the sum of an infinite series? What suggestions would you have for someone new to learning them? Memorize identities? Also, what are the best resources for learning about them? I have searched for resources on Google, but they all give different examples with different ways to arrive at answers which have further confused me.

2 Answers

  1. Daniel- Reply

    2019-11-14

    Are there any hard and fast rules for finding the sum of an infinite series?

    I guess it depends on what you mean by hard and fast rules. But if my guess is right, I would say no. That aren't any standard procedure or rules that will allow you to find the sum of any convergent series.

    What suggestions would you have for someone new to learning them? Memorize identities? Also, what are the best resources for learning about them?

    I would suggest you, as is done in the other answer, to study this from a good calculus book. I would agree that Stewart's calculus book would be a good start (just my opinion). One thing that is done there is to realize that a lot of functions can be written as power series. So you for example have $$ e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!} $$ for all $x$. So if you ever want to compute the sum of the series $$\sum_{n=0}^{\infty} \frac{1}{n!} $$ you realize that this is just $e^1 = e$. Now of course this will only allow you to find sums of some types of series, but it is a start. So would I recommend that you memorize identities? Yes, to some extent I would. To use formulas you usually have to remember them. I can understand if you have a desire to have a good intuitive understanding of series, but if you are just starting out, then just try and get some basic stuff down. Even though Google is your friend, it (IMO) can be a good idea to just pick a text book and follow a coherent treatment given instead of trying to put all the pieces together yourself. As you study make sure that you don't just look at the theory, but make sure that you do a lot of exercises. Don't just study solutions to problem, but try to solve problems yourself.

  2. Dennis- Reply

    2019-11-14

    Well, if you're new to calculus, I'd take a look at Stewart's Calculus first. There's a section on infinite sums and series. If that's too easy for you, maybe look at Taylor and Mann's Advanced Calculus. These are decent places to start. It's probably best to just get familiar with infinite series in general.

    Though keep in mind, different people's learning preferences always differ.

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