Index shift

In the case of sums or series, index shift or index shift describes the substitution of the summation index by adding an integer. The value of the sum or row itself does not change. The purpose of such an index shift is usually to simplify the further calculation.

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

example

${\displaystyle \sum _{k=1}^{4}(k+4)=(1+4)+(2+4)+(3+4)+(4+4)=26}$

If you replace , you get for the summands ${\displaystyle t=k+4}$

${\displaystyle t=(k+4)}$,

for the first index of the sum

${\displaystyle k=1\Leftrightarrow t-4=1\Leftrightarrow t=5}$

and for the last index of the sum

${\displaystyle k=4\Leftrightarrow t-4=4\Leftrightarrow t=8}$.

This gives you the new totals display

${\displaystyle \sum _{t=5}^{8}t=5+6+7+8=26}$.

Index shift in other operations

Index shifts can be used in an analogous manner in other mathematical operations that have a continuous index. The same applies analogously to a product

${\displaystyle \prod _{k=m}^{n}a_{k}=\prod _{k=m+z}^{n+z}a_{k-z}.}$

swell

• Calculus Textbook - Part 1 . 1980; 17th edition, Teubner 2009, ISBN 978-3835101319
• Textbook of Analysis - Part 2 . 1981; 14th edition, Teubner 2008, ISBN 978-3835102088