Discrete Math Problem Find a formula for (1 / (1 · 2)) + (1 / (2 · 3)) + (1 / (3 · 4)) + . . . + (1 /(n(n + 1) )

Question

Find a formula for (1 / (1 · 2)) + (1 / (2 · 3)) + (1 / (3 · 4)) + . . . + (1 /(n(n + 1) ) by examining the values of this expression for small values of n, where n is a positive integer. Use mathematical induction to prove your result.

I know there is a similar problem solved, but I am looking for all positive integers, rather than just integers greater than 2.

Hint: The closed form of the sum is $1-\frac 1{n+1}$. Try to prove this using induction on your own (although this is trivial by a telescoping sum argument). — learner, Apr 14 '16 at 23:37
See also: What is the formula for $\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\cdots +\frac{1}{n(n+1)}$ — Martin Sleziak, Jun 18 '22 at 10:41

score 1 · Answer 1 · answered Apr 15 '16 at 01:22

1

We know that $\frac{1}{i(i+1)} = \frac{1}{i} - \frac{1}{i+1}$

So the solution would be $$ \frac{1}{1\cdot 2} + \cdots + \frac{1}{n(n+1)} = 1 - \frac12 + \frac12 - \frac23 + \cdots -\frac{1}{n} +\frac{1}{n} -\frac{1}{n+1} = \frac{n}{n+1} $$

answered Apr 15 '16 at 01:22

Serina

201

Discrete Math Problem Find a formula for (1 / (1 · 2)) + (1 / (2 · 3)) + (1 / (3 · 4)) + . . . + (1 /(n(n + 1) )

1 Answers1