# How do you evaluate the sum represented by sum_(n=1)^5n/(2n+1) ?

Oct 13, 2014

First expand the series for each value of n $\frac{n}{2 n + 1}$=1/(2(1)+1+2/(2(2)+1+3/(2(3)+1+4/(2(4)+1+5/(2(5)+1

Next, perform the operations in the denominator...

$\frac{1}{3}$+$\frac{2}{5}$+$\frac{3}{7}$+$\frac{4}{9}$+$\frac{5}{11}$

Now, to add fractions we need a common denominator... in this case it's $3465$

Next, we have to multiply each numerator and denominator by the missing components...

$\frac{1}{3}$ gets multiplied by $1155$ giving $\frac{1155}{3465}$

(Divide the $3465$ by $3$ to get $1155$ and divide the rest by the given denominator.)

2/5*693/693=1386/3465, 3/7*495/495=1485/3465, 4/9*385/385=1540/3465 and 5/11*315/315=1575/3465

Now simply add the numerators together... $\frac{1155 + 1386 + 1485 + 1540 + 1575}{3465}$

giving $\frac{7141}{3465}$.