Science:Infinite Series Module/Units/Unit 1/1.2 Sigma Notation/1.2.05 Changing Summation Limits Example

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Problem

Change the following summation

Failed to parse (unknown function "\begin{align}"): {\displaystyle \begin{align}  \sum_{k=3}^{\infty} \frac{k}{2+k} = \frac{3}{2+3}+\frac{4}{2+4}+\frac{5}{2+5}+\ldots , \quad (1)\end{align} }


so that the index of summation start at 1 instead of at 3.

Complete Solution

Although not necessary, we will use two method for solving this problem: Method 1 and Method 2.

Method 1

Introducing the transformation


so that when  gives us

as desired.

Method 2

Our procedure is to add and subtract terms in the sum to shift our index to 1:

Failed to parse (unknown function "\begin{align}"): {\displaystyle \begin{align} \sum_{k=3}^{\infty} \frac{k}{2+k} &= \frac{3}{2+3}+\frac{4}{2+4}+\frac{5}{2+5}+\ldots && (2)  \\ \\ &= \Big( \mathbf{\color{Purple}{ \frac{2}{2+2} }} - \frac{2}{2+2} \Big) + \frac{3}{2+3}+\frac{4}{2+4}+\frac{5}{2+5}+\ldots && (3) \\ \\ &= \Big( - \frac{2}{2+2} \Big) + \mathbf{\color{Purple}{ \frac{2}{2+2} }} + \frac{3}{2+3}+\frac{4}{2+4}+\frac{5}{2+5}+\ldots && (4) \\ \\ &= \Big( \mathbf{\color{Mahogany}{ \frac{1}{2+1} }} - \frac{1}{2+1} \Big) + \frac{1}{2+1} + \mathbf{\color{Purple}{ \frac{2}{2+2} }} + \frac{3}{2+3}+\frac{4}{2+4}+\frac{5}{2+5}+\ldots && (5) \\ \\ &= \Big( - \frac{1}{2+1} \Big) + \mathbf{\color{Mahogany}{ \frac{1}{2+1} }} +\frac{2}{2+2} + \frac{3}{2+3}+\frac{4}{2+4}+\frac{5}{2+5}+\ldots && (6) \\ \\ &= \Big( - \frac{1}{2+1} - \frac{2}{2+2} \Big) + \sum_{k=1}^{\infty} \frac{k}{2+k} && (7) \\ \\ &= - \frac{5}{6} + \sum_{k=1}^{\infty} \frac{k}{2+k} && (8) \\ \\ \end{align}}

as desired. Therefore

Discussion of Some Steps

Method 1

The transformation , was chosen to that the index would start at 1.

Method 2

Most steps in this approach involved straightforward algebraic manipulation. Steps (3) and (5) involve adding and subtracting terms in a way that will allow us to change the summation limits.

More precicesly, in Step (3) we added and substracted the term, and in Step (5) we added and subtracted the term.

Potential Challenges

Method 2 Requires More Work, So Why Should I Use It?

You may not have a choice. In some circumstances, you need to convert your sum in a way that does not change the general term.