I am currently studying sequence which I think will lead up to my next topic induction.
My question is if $$\sum_{k=0}^n \frac{k+1}{n+k}= \frac{1}{n}+\frac{2}{n+1}+\frac{3}{n+2}+\cdots+\frac{n+1}{2n}$$
how is this summation notation $$\sum_{k=0}^n\frac{k+1}{n+k}$$ equal to this expanded form $\frac{n+1}{2n}$
They give different values so how is it the summation notation of the expanded form ?
EDIT: The reason for my question comes from Discrete Algebra and its Application Taken from Discrete Algebra and its Application
There seems to be a confusion as how you read the question you link. It asks you to find the summation notation for $$ \frac{1}{n}+\frac{2}{n+1}+\frac{3}{n+2}+\cdots+\frac{n+1}{2n} \tag{1}. $$ $(1)$ is then called expanded form of the sum, because... well, it lists all the terms that are summed.
Then, the answer says that the summation notation for the expanded form $(1)$ is $$\sum_{k=0}^n \frac{k+1}{n+k} \tag{2} $$ that is, $(2)$ is a shorter, condensed rewriting of $(1)$. That is all that is claimed: $(1)=(2)$, i.e. the two are equal: $$ \sum_{k=0}^n \frac{k+1}{n+k}= \frac{1}{n}+\frac{2}{n+1}+\frac{3}{n+2}+\cdots+\frac{n+1}{2n}. $$ Nothing more.