Math Is Fun Forum

EMPhillips1989

Member

Registered: 2008-01-21

Posts: 40

sequences

hey can anyone help me prove that the sequence

converges to 2??

any advise would be much appreciated!!!

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: sequences

Consider the sum of the first k+1 terms:
S := 1 + 1/2 + 1/4 + ... + 1/(2^k)

Now look at 2S.
This is 2 + 1 + 1/2 + ... + 1/(2^(k-1)).

Taking S away from 2S gives S = 2 - 1/(2^k).
1/(2^k) goes to 0 as k gets large, and you're left with 2.

---

Why did the vector cross the road?
It wanted to be normal.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: sequences

Look up the phrase "geometric series" on google.

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Re: sequences

S = 1 + 1/2 + 1/4 + 1/8 + ...
S/2 = 1/2 + 1/4 + 1/8 + 1/16 + ...

S - S/2 = 1
S/2 = 1
S = 2

Last edited by Daniel123 (2008-02-11 11:34:26)

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: sequences

The series is geometric with first term 1 and common ratio ½; hence the sum is