Mathematical Induction problem.

[charmedbeauty]

Homework Statement

(1/2!)+(2/3!)+(3/4!)+...+(n/(n+1)!)

a) calculate for a few small values of n.
b) Make a conjecture about a formula for this expression
c)Prove your conjecture by mathematical induction.

Homework Equations

The Attempt at a Solution

So for the first part I just used values n=1,2,3

so..

n=1

1/2!=1/2

n=2

2/3! = 1/6+1/2

n=3

3/4! = 1/8+1/6+1/2.

for part b)

make a conjecture about the formula

It should be ((n+1)!-1)/(n+1)!

for part c) I am getting stuck...

test for n=1, which is true

assume true for n=k

show true for n=k+1

so...

(1/2!)+(2/3!+(3/4!)+...+(k/(k+1)!)+((k+1)/(k+2)!) = ((k+2)!-1)/(k+2)!

where

(1/2!)+(2/3!+(3/4!)+...+(k/(k+1)!)= ((k+1)!-1)/(k+1)!

so...

((k+1)!-1)/(k+1)!+(k+1)/(k+2)!= ((k+2)!-1)/(k+2)!

but I am lost as to where to go from here, have I made a mistake?
Help!

 

[Office_Shredder]

2/3! is 1/3, not 1/6. Witing it out as 1/8+1/3+1/2 (correcting your answer for n=3) is not particularly enlightening. Put the fractions together!

n=1: 1/2
n=2: 5/6
n=3 47/48

is some sort of pattern beginning to emerge?

Your conjectured formula is just the definition so isn't what they're looking for. What you should try to do is come up with what's called a "closed form" expression - an expression for the sum that involves no big summation of terms.

 

[charmedbeauty]

2/3! is 1/3, not 1/6. Witing it out as 1/8+1/3+1/2 (correcting your answer for n=3) is not particularly enlightening. Put the fractions together!

n=1: 1/2
n=2: 5/6
n=3 47/48

is some sort of pattern beginning to emerge?

Your conjectured formula is just the definition so isn't what they're looking for. What you should try to do is come up with what's called a "closed form" expression - an expression for the sum that involves no big summation of terms.

How about now is that correct for b)??(look at original post I edited). but I'm a little lost on c)