The series 1 k(k+1) k = 1 can be rewritten in the form 1 k − 1 k+1 k = 1 using partial fraction decomposition. a) Find the fourth partial sum S4 b) Find the nth partial sum Sn

Question

The series    1 k(k+1)   k = 1  can be rewritten in the form    1 k  −  1 k+1   k = 1  using partial fraction decomposition.   a) Find the fourth partial sum S4        b) Find the nth partial sum Sn

anonymous · Answer

Here what we have
$$
\sum _{k=1}^n \frac{1}{k (k+1)}=\sum _{k=1}^n
   \left(\frac{1}{k}-\frac{1}{k+1}\right)
$$

anonymous · Answer

$$

S_4=\sum _{k=1}^4 \left(\frac{1}{k}-\frac{1}{k+1}\right)=\
1-\frac 1 2+\
\frac 1 2 -\frac 13 +\
\frac 1 3 -\frac 14 +\
\frac 1 4-\frac 15  \=
1-\frac 1 5=\frac 4 5

$$
By telescopping

anonymous · Answer

$$
S_n=\sum _{k=1}^n \left(\frac{1}{k}-\frac{1}{k+1}\right)=
1-\frac 1 {n+1}


$$

All terms geta nnihilated except 1 and$\large -\frac{ 1} {n+1}$