Criterion for convergence of positive series

The criterion for the convergence of positive series is the main sign of the convergence of positive number series . Argues that the positive row $\sum _{k=1}^{\infty }a_{k}$ ${\ displaystyle \ sum _ {k = 1} ^ {\ infty} a_ {k}}$ $\ sum _ {{k = 1}} ^ {\ infty} a_ {k}$ converges if and only if the sequence of its partial sums $S(n)=\sum _{k=1}^{n}a_{k}$ ${\ displaystyle S (n) = \ sum _ {k = 1} ^ {n} a_ {k}}$ $S (n) = \ sum _ {{k = 1}} ^ {n} a_ {k}$ limited from above.

Proof

On the one hand, since the series converges, the sequence of partial sums has a limit. Therefore, it is limited. So it is limited both from below and from above.

Conversely, let a positive series be given and a sequence of partial sums bounded above. Note that the sequence of partial sums is non-decreasing:

S_{n+1}-S_{n}=a_{n+1}\geqslant 0

{\ displaystyle S_ {n + 1} -S_ {n} = a_ {n + 1} \ geqslant 0}

Now we use the property from the monotonic sequence theorem . We find that the sequence of partial sums converges (it does not monotonically decrease and is bounded above), and therefore the series converges by definition.

Literature

Yu. S. Bogdanov - “Lectures on mathematical analysis” - Part 2 - Minsk - Publishing house of BSU named after V.I. Lenin - 1978.

Criterion for convergence of positive series

Proof

Literature

More articles: