Question

Use mathematical induction to show
1/2 + 1/4 + ... 1/(2^n) = 1 - 1/(2^(n+1))

Answer 1

Are you sure you typed it right? Normally with induction, we show that it balances for n=1, then your induction hypothesis ASSUMES it works for n= n. Then you show that it works for the (n + 1)th case.

When n= 1, left side becomes 1/2
Right side = 1 - 1/(2^2) = 1-1/4 = 3/4
1/2 doesn't equal 3/4
If it doesn't even work for the first case, I'm not sure how you'll prove it.

Answer 2

Use mathematical induction to show
1/2 + 1/4 + ... 1/(2^n) = 1 - 1/(2^(n)) ***

sorry I typed out half of the original and then half of my notes on the n+1 case

Answer 3

Okay. Now it makes sense.
So you first show that it is true when n=1:
1/2 = 1- (1/2) = 1/2
Now say: Induction Hypothesis:
Case where n = k Let us assume that: 1/2 + 1/4 + ... + 1/(2^k) = 1 - 1/(2^k)
Now when n = k+1, the left hand side becomes
1/2 + 1/4 +...+1/k + 1/(k+1)
1/2 + 1/4 + ...+1/k = 1-1/(2^k) from our I.H. (Induction Hypothesis) above.
Add 1/(k+1) to that.
We're almost done.

Answer 4

I'm going to write it on paper, scan it and attach the file to make it clearer. One minute.

Answer 5

Thank you!

Answer 6

Thank you!

Answer 7

Also would you happen to know the sub-sequential limits for sn = sin(npi/4)

Answer 8

No sorry.

Answer 9

Thank you for your help. I appreciate it.

Answer 10

No problem. Post the other question. Maybe someone else can help.