Question

Write the following sum using summation notation.

$\frac{-2}{5}+\frac{3}{6}-\frac{4}{7}+...+\frac{11}{14}$

Short answer

Summation notation of the given sum is$\sum _{k=1}^{10}\left[{\left(-1\right)}^k\left(\frac{k+1}{k+4}\right)\right]$.

Step-by-step solution

Given information:

Consider the given sum,$\frac{-2}{5}+\frac{3}{6}-\frac{4}{7}+...+\frac{11}{14}$

Expanding the sum:

$\frac{-2}{5}+\frac{3}{6}-\frac{4}{7}+\frac{5}{8}-\frac{6}{9}+\frac{7}{10}-\frac{8}{11}+\frac{9}{12}-\frac{10}{13}+\frac{11}{14}$

Here,$k=1to10$

Step 3: 

(1) Since the signs in the given sum is alternate, ${\left(-1\right)}^k$.

(2) Numerator :$2,3,4,...,11$

The above sequence is one increased by the previous, $ie,k+1$.

(3) Denominator : $5,6,7,...,14$

The above sequence is four increased by the previous, $ie,k+4.$

Summing up 3 points, we get,

$\sum _{k=1}^{10}\left[{\left(-1\right)}^k\left(\frac{k+1}{k+4}\right)\right]$