(Changing) Limits of a Summation

[martina1075]

Homework Statement:

Good afternoon 😃 Can someone explain to me how to change the bounds of summation using this example please?

Relevant Equations:

As per picture

 

[fresh_42]

What does $\sum_{i=k}^m a_i$ stand for?

 

[PeroK]

Try $j = i - m + p$.

 

[martina1075]

What does $\sum_{i=k}^m a_i$ stand for?

i stands for the positive integer , ai stands for a unique number and m and n are positive integers with m being greater or equal to n.

 

[fresh_42]

i stands for the positive integer , ai stands for a unique number and m and n are positive integers with m being greater or equal to n.

No, that's what the "numbers" mean. But what is the abbreviation $\sum$ for?

 

Write both sides out with the definition of the summation symbol. Shouldn't be too hard to see thay are the same.

 

[SammyS]

What does $\sum_{i=k}^m a_i$ stand for?

i stands for the positive integer , a_i stands for a unique number and m and n are positive integers with m being greater or equal to n.

(Notice that fresh has changed some of the variables from those in the problem statement.)

Think of what @fresh_42 was getting at more like:
$\displaystyle \sum_{i=m}^n a_i$ stands for:

$a_{m} + a_{m+1} + a_{m+2} + ... + a_{n-1} + a_{n}$​

In words, sum the $a_i$ values, where $i$ takes on values from $m$ through $n$.

With this in mind, do a similar translation for the right hand side. It may help to use an index other than $i$, initially.

$\displaystyle \sum_{j=p}^{p+n-m} a_{j+m-p}$

Plug $p$ in for $j$ to get the index for the first term in the sum, then plug $p+n-m$ in for $j$ to get the index for the final term.

(Added in Edit: Fixed summation per @archaic's comment in the next post.)

 

[archaic]

(Notice that fresh has changed some of the variables from those in the problem statement.)

Think of what @fresh_42 was getting at more like:
$\displaystyle \sum_{i=m}^n a_i$ stands for:

$a_{m} + a_{m+1} + a_{m+2} + ... + a_{n-1} + a_{n}$​

In words, sum the $a_i$ values, where $i$ takes on values from $m$ through $n$.

With this in mind, do a similar translation for the right hand side. It may help to use an index other than $i$, initially.

$\displaystyle \sum_{i=p}^{p+n-m} a_{j+m-p}$

Plug $p$ in for $j$ to get the index for the first term in the sum, then plug $p+n-m$ in for $j$ to get the index for the final term.

You forgot to change $i$ to $j$ in the second symbol :)