convergent series

With the norm in place, absolutely convergent series can be used instead of finite sums.

By taking we normally recover the usual summation for convergent series.

For absolutely convergent series, this theorem, albeit true, is almost evident.

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For this we compare with the convergent series .

There he wrote papers on convergent series, interpolation theory, and number theory.

Let x be an absolutely convergent series in X.

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By extension, any convergent series would work.

Therefore, the series converges as the difference of two convergent series .

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Analogously of course, the matrices whose row sums are absolutely convergent series also form a ring.

An example of a conditionally convergent series is the alternating harmonic series.

A convergent series of numbers can often be reordered in such a way that the new series diverges.

One way to do this is by means of a convergent series of inverted rising exponentials.

A summability method M is regular if it agrees with the actual limit on all convergent series.

In consequence, the characteristic function of the log-normal distribution cannot be represented as an infinite convergent series.

The reciprocals of factorials produce a convergent series: (see e)

The reciprocals of square numbers produce a convergent series (the Basel problem):

Unfortunately the corresponding convergent series converges very slowly.

Hans Rademacher, in 1937, was able to refine their formula to find an exact convergent series solution to this problem.

The path formed by connecting the partial sums of a conditionally convergent series is infinitely long.

The general principle is that addition of infinite sums is only commutative for absolutely convergent series.

Note the remarkable similarity to the globally convergent series expression for the Hurwitz zeta function.

Every absolutely convergent series is unconditionally convergent, but the converse implication does not hold in general.