Q. 14 - Calculus

Question

The algebra of sums: Fill in the blanks to complete the sum rules that follow. You may assume that $a_k$ and $b_k$are functions defined for nonnegative integers $k$ and that $c$ is any real number.

$\sum _{k=m}^{p-1}{a}_k+\sum _{k=p}^n{a}_k=$______

Step-by-Step Solution

Answer

$\sum _{k=m}^{p-1}{a}_k+\sum _{k=p}^n{a}_k=\sum _{k=m}^n{a}_k$.

Step 1 . Given information

$\sum _{k=m}^{p-1}{a}_k+\sum _{k=p}^n{a}_k$.

Step 2 . First write the sigma notation in the expanded form.

$$\begin{gathered} \sum _{k=m}^{p-1}{a}_k+\sum _{k=p}^n{a}_k=\left(a_m+a_{m+1}+\cdots +a_{p-1}\right)+\left(a_p+a_{p+1}+\cdots +a_{n-1}+a_n\right) \\ =a_m+a_{m+1}+\cdots +a_{p-1}+a_p+a_{p+1}+\cdots +a_{n-1}+a_n \end{gathered}$$

Now write the obtained sequence in sigma notation.

$a_m+a_{m+1}+\cdots +a_{p-1}+a_p+a_{p+1}+\cdots +a_{n-1}+a_n=\sum _{k=m}^n{a}_k$.

where $m,n,p$ are any integers such that $0\le m<p<n$.