The rest of the row

A series obtained by dropping from the original n first terms is called the nth remainder of the series .

Designation:

r_{n}=\sum _{k=n+1}^{\infty }a_{k}

{\ displaystyle r_ {n} = \ sum _ {k = n + 1} ^ {\ infty} a_ {k}}

{\ displaystyle r_ {n} = \ sum _ {k = n + 1} ^ {\ infty} a_ {k}}

All members, except those that are in the nth remainder of the series, add up to the so-called. nth partial sum of the series .

Properties

For the remainder of the series the following statements are true:

If the series converges , then any remainder of it converges.
If at least one remainder of the series converges, then the series itself also converges.
If the series converges, then

\lim _{n\to \infty }\sum _{k=n+1}^{\infty }a_{k}=0

{\ displaystyle \ lim _ {n \ to \ infty} \ sum _ {k = n + 1} ^ {\ infty} a_ {k} = 0}

There are methods for estimating the remainder of a series using the Cauchy integral criterion (for a sign-positive series) and the Leibniz Convergence Sign (for an alternating series ).

The rest of the row

Properties

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