sigma notation

Question

explain why

99
Σ (j-1)/(j+1)
j=0


equals

100
Σ (j-2)/j
j=1

Answer 1

A little "mathematically", 

₉₉
Σ (j-1)/(j+1)=|denote i=j+1, then j-1=i-2, if j=0, i=1;if j=99,i=100|=
^_j=0

(now by using i as the summation indicator)

   ₁₀₀
=Σ (i-2)/i.  No matter what is the index: i ,j or something else. 
  ⁱ⁼¹

If you want, you can replace i by j again,  to have 

  _¹⁰⁰
=Σ (j-2)/j. 
 ^j=1

 

Answer 2

Write out the first three or four terms in each of the Summations.

Write out the last three or four terms in each of the Summations.

What do you see?