[Math] the difference between convergence of a sequence and convergence of a series

calculusconvergence-divergencesequences-and-series

I'm preparing for my calculus exam and I'm unsure how to approach the question: "Explain the difference between convergence of a sequence and convergence of a series?"

I understand the following:

Let the sequence $a_n$ exist such that $a_n =\frac{1}{n^2}$

Then $\lim_{n\to\infty} a_n=\lim_{n\to\infty} \frac{1}{n^2}=0$ therefore $a_n$ converges to $0$.

And the series $\sum_{i=1}^{n}a_n=1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + … +\frac{1}{n^2}$

And by the $n$-th term test, this series converges. But, I don't understand why or how the convergence between the series and the sequence is different.

I looked online and I find a lot of answers on how to determine convergence or divergence, but the only difference I've found is that you use limits to test sequences and series have more complex testing requirements. Please help!

Best Answer

Series can be baffling things. The trouble is that the convergence of the terms tells you nothing about the convergence of the series. We know that $\lim_{n\to\infty}a_n=0$ is necessary for a series $\sum_{k=0}^\infty a_n$ to converge, but it is not sufficient.

For example consider the sequences $a_n=1/n$, $b_n=(-1)^n/n$, $c_n=1/n^2$, $d_n=1/n^3$, $e_n=1/n^5$. All of these sequences converge to zero. But:

  • $\sum_{k=1}^\infty a_n$ diverges (harmonic series)
  • $\sum_{k=1}^\infty b_n$ converges (alternating series)
  • $\sum_{k=1}^\infty c_n$ converges to $\pi^2/6$
  • $\sum_{k=1}^\infty d_n$ converges, to an irrational number (ApĂ©ry's theorem)
  • $\sum_{k=1}^\infty e_n$ converges, but its not known if the result is rational or irrational.

There is, as far as I know, no complete decision method for convergence of a series. You try a sequence of tests, and each one will return a result of converge, diverge, undecided. If the latter, you try another test.

There are certain families of series, such as geometric series and $p$-series, for which convergence or divergence is trivial ($\sum_{k=0}^\infty r^k$ converges if and only if $|r|<1$), but in general, given a new series which doesn't fit into a known class, you're on your own.

As far as I know, this problem is unsolved: is there a sequence of rational numbers $a_n$ for which $\lim_{n\to\infty}a_{n+1}/a_n=0$ and $\sum_{n=0}^\infty a_n=\pi$? (Note that if we replace $\pi$ by $e$ then the result is trivial, as we can put $a_n=1/n!$.)

As you see, series are tricky.

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