Question

Can someone help me find the partial sum of:

Answer 1

\[\frac{ 1 }{n(n+1)}\] that's the term of the series

Answer 2

The series begins at n=1

Answer 3

and end up at... ?

Answer 4

Maybe at "n" @Diogo

Answer 5

then just put a sigma from n=1 to +oo before

Answer 6

It ends up at infinity

Answer 7

\[\sum_{n=1}^{\infty} \frac{ 1 }{ n (n+1) }\]

Answer 8

partial sums may form patterns that can be defined explicitly; otherwise it might be best to do a partial decomp and try to do a term by term approach

Answer 9

isn't there an other way?

Answer 10

im sure there are plenty of methods out there ...

Answer 11

\[\frac12=\frac12\\
\frac12+\frac16=\frac{8}{12}\\
\frac23+\frac1{12}=\frac{27}{36}\\\]
etc ..

Answer 12

see if the bottom and top can be defined as sequences

Answer 13

\[\sum_{n=1}^{\infty} \frac{ 1 }{ n(n+1) } = 1- \frac{ 1 }{\infty } = 1 - 0 =\]

Answer 14

\[S_n=\frac{n}{n+1}\]maybe

Answer 15

Actually I don't understand what Diogo wrote...

Answer 16

Diogo was interpreting this as the ultimate partial sum, the TOTAL sum as n goes to infinity

Answer 17

the partial sums are the intermediary sums from 1 to n, for a given integer n

Answer 18

How does he get to the final value then?

Answer 19

he has his ways :)

Answer 20

but the question, as far as i can tell, is not asking for the SUM of the series as a whole, but rather as an explicit rule in terms of n. as n approaches infinity, the rule simplifies to 1

Answer 21

30 minutes to a final .. ive got to get a little refresher studying in

Answer 22

I think I got it! so you make the partial sum of the first few numbers and then you try to recognize a pattern. thanks

Answer 23

\[ \frac{ 1 }{ n(n+1) } = \frac{ 1+n -n }{ n(n+1) } = \frac{ n+1 }{ n(n+1) } - \frac{ n }{ n(n+1) } = \frac{ 1 }{ n } - \frac{ 1 }{ (n+1) } \]


for n = 1 you have

1-0.5

for n = 2 you have

0.5-0.3333

for n=3 you have

0.3333-0.25


If you sum all you'll have

1-0.5 + 0.5 - 0.3333 +0.3333-0.25 +0.25 + ....

as you can see all the terms cancel each other so the final answer is just 1

Answer 24

ok thanks