Expert: Paul Klarreich - 9/17/2009

Question
100
SUM ( 1/(i+4) -  1/(i+5) )
i=25

please help me solve this question. I am confused since there is a fraction in the function relation.

Answer
Questioner: Andrew
Country: Ghana
Category: Calculus
Private: No
Subject: Calculus (Summation)
Question: 100
SUM ( 1/(i+4) -  1/(i+5) )
i=25

please help me solve this question. I am confused since there is a fraction in the function relation.
..................................................
Hi, Andrew,

First things:  Be VERY careful about use of vocabulary in mathematics.  If you don't use the vocabulary correctly, you won't know what the words mean.  
If you don't what the words mean, you won't know what to do.

I think you want to

EVALUATE THE SUM:

100
SUM   ( 1/(i+4) -  1/(i+5) )
i=25

Now that's out of the way:

This is a 'telescoping sum'.  If you try writing it out, you will see:
1    1
-- - ---     << i = 25
29   30

1    1
-- - ---     << i = 26
30   31

1    1
-- - ---     << i = 27
31   32

...
...

1    1
-- - ---     << i = 99
103  104

1    1
-- - ---     << i = 100
104  105

================== << SUMMING UP

Now observe that ALMOST every term appears as + and -, so they cancel.
EXCEPT:

1    1
-- - ---
29   105

And that is your sum.