0
$\begingroup$

Recently, I was working on Principles of real analysis by Rudin. When I got into the topic of convergence of series (Ch3), the book said if partial sum of the series converge, then the series converge.

And, an idea popped into my head that if there is a systematic way to find out the formula of the partial sum, like geometric series does, so I can test the convergence of the series simply by showing that the limit of the formula and does not need to use so many tests, like comparison test, ratio test, root test, etc.

$\endgroup$
  • $\begingroup$ This is no easy task, bro. This is the definition of a series, isn't it? $\endgroup$ – Shoutre Nov 20 '15 at 3:47
  • 1
    $\begingroup$ Sometimes there is, in general there isn't. When there is, then yes of course you can try to reason about the expression that you manipulate the sum into. $\endgroup$ – BrianO Nov 20 '15 at 3:48
  • $\begingroup$ Can you find a simple exact formula for $\displaystyle\sum_{k=1}^n\frac1k$? $\endgroup$ – David Nov 20 '15 at 4:01
0
$\begingroup$

Finding a closed form for the partial sums of a sequence is a much more difficult problem than merely testing the convergence, or even finding a value for the infinite sum if it does converge. In fact, finding the exact value of an infinite sum is usually no small matter. See here: https://en.wikipedia.org/wiki/Basel_problem

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.