# Telescope sum

Pushing together a telescope - namesake of the telescope sum
A collapsible telescope

A telescoping series is in the mathematics a finite sum of differences (except the first and the last) cancel each other, wherein each two neighboring members. This process is called telescoping a sum . The term is derived from pushing two or more cylindrical tubes into one another .

If there is a consequence , then is ${\ displaystyle (a_ {k}) _ {k \ in \ mathbb {N}} \,}$

${\ displaystyle \ sum _ {i = 1} ^ {n} (a_ {i} -a_ {i + 1})}$

a telescope sum. If you can write a sum as a telescope sum, its evaluation is simplified:

${\ displaystyle \ sum _ {i = 1} ^ {n} (a_ {i} -a_ {i + 1}) = (a_ {1} -a_ {2}) + (a_ {2} -a_ {3 }) + \ cdots + (a_ {n-1} -a_ {n}) + (a_ {n} -a_ {n + 1}) = a_ {1} -a_ {n + 1}}$.

A series whose partial sums are telescope sums is called a telescope series . A row of telescopes

${\ displaystyle \ sum _ {i = 1} ^ {\ infty} (a_ {i} -a_ {i + 1})}$

is convergent if and only if converges to a limit value . The sum of the series is then the same . ${\ displaystyle (a_ {k}) _ {k \ in \ mathbb {N}} \,}$${\ displaystyle g \,}$${\ displaystyle a_ {1} -g \,}$

The same applies to a product:

${\ displaystyle \ prod _ {i = 1} ^ {n} {\ frac {a_ {i + 1}} {a_ {i}}} = {\ frac {a_ {2}} {a_ {1}}} \ cdot {\ frac {a_ {3}} {a_ {2}}} \ cdot {\ frac {a_ {4}} {a_ {3}}} \ cdots {\ frac {a_ {n}} {a_ { n-1}}} \ cdot {\ frac {a_ {n + 1}} {a_ {n}}} = {\ frac {a_ {n + 1}} {a_ {1}}}}$

is a telescope product , so to speak .

The situation is more complicated when the telescope runs over three (or more) consecutive links (see example).

In number theory telescopic sums represent an important tool.

## Examples

${\ displaystyle (x-1) \ sum _ {k = 0} ^ {n} x ^ {k} = \ sum _ {k = 0} ^ {n} (x ^ {k + 1} -x ^ { k}) = x ^ {n + 1} -1.}$
The partial fraction decomposition of can be obtained using, for example${\ displaystyle {\ frac {1} {k (k + 1)}}}$
${\ displaystyle {\ frac {1} {k (k + 1)}} = {\ frac {k + 1-k} {k (k + 1)}} = {\ frac {k + 1} {k ( k + 1)}} - {\ frac {k} {k (k + 1)}} = {\ frac {1} {k}} - {\ frac {1} {k + 1}}}$.
It follows
${\ displaystyle \ sum _ {k = 1} ^ {n} {\ frac {1} {k (k + 1)}} = \ sum _ {k = 1} ^ {n} \ left ({\ frac { 1} {k}} - {\ frac {1} {k + 1}} \ right) = 1 - {\ frac {1} {n + 1}}.}$
• three times the telescope sum:
{\ displaystyle {\ begin {aligned} (x-1) ^ {2} \ sum _ {k = 1} ^ {n} kx ^ {k-1} & = \ sum _ {k = 1} ^ {n } (kx ^ {k + 1} -2kx ^ {k} + kx ^ {k-1}) = \ sum _ {k = 1} ^ {n} [kx ^ {k + 1} - (k-1 ) x ^ {k}] - \ sum _ {k = 1} ^ {n} [(k + 1) x ^ {k} -kx ^ {k-1}] \\ & = nx ^ {n + 1 } - (n + 1) x ^ {n} +1. \ end {aligned}}}
Alternatively, this follows for by differentiation from the first example with the help of the quotient rule :${\ displaystyle x \ neq 1}$
${\ displaystyle \ sum _ {k = 1} ^ {n} kx ^ {k-1} = {\ frac {\ rm {d}} {{\ rm {d}} x}} \ sum _ {k = 0} ^ {n} x ^ {k} = {\ frac {\ rm {d}} {{\ rm {d}} x}} {\ frac {x ^ {n + 1} -1} {x- 1}} = {\ frac {(n + 1) x ^ {n} (x-1) - (x ^ {n + 1} -1)} {(x-1) ^ {2}}} = { \ frac {nx ^ {n + 1} - (n + 1) x ^ {n} +1} {(x-1) ^ {2}}}}$.
This is an important application example of differential calculus as a calculus for term transformation .

## literature

• Rolf Walter: Introduction to Analysis. Volume 1. Walter de Gruyter, Berlin et al. 2007, ISBN 978-3-11-019539-2 , pp. 31–32 ( excerpt in the Google book search).
• Harro Heuser : Textbook of Analysis. Volume 1. 6th edition, unchanged reprint of the 5th revised edition. Teubner, Stuttgart et al. 1989, ISBN 3-519-42221-2 , pp. 91, 94, 194.

• Philippe B. Laval: Telescoping Sums (PDF; 95.1 kB) - Derivation of a general theorem for telescoping sums of the form${\ displaystyle \ sum _ {n = 1} ^ {\ infty} {\ frac {1} {n \ cdot (n + k)}}}$