If $M= \displaystyle \frac{1}{90\cdot 46}+\frac{1}{89\cdot 47}+\cdots \cdots \cdots ++\frac{1}{46\cdot 90}$
and $\displaystyle N = 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots \cdots -\frac{1}{90}.$ Then $M/N$ is
Plan $$N=\bigg(1+\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{90}\bigg)-2\bigg(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\cdots \bigg)=\sum^{45}_{k=1}\frac{1}{45+k}.$$
How do i convert $M$ into $N$ form . Help me please
By your work: $$M=\sum_{k=1}^{45}\frac{1}{(91-k)(k+45)}=\frac{1}{136}\sum_{k=1}^{45}\left(\frac{1}{91-k}+\frac{1}{k+45}\right)=\frac{1}{68}\sum_{k=1}^{45}\frac{1}{k+45}=\frac{1}{68}N.$$