Year 12s: what will be the first way you research your future options? >>

Year 12s: what will be the first way you research your future options? >>

Year 12s: what will be the first way you research your future options? >>

whats this called?

thanks

(edited 13 years ago)

Daniel Freedman

I don't think it has a special name.... Just a series.

Why? Are you trying to sum it?

OP

Original post by Daniel Freedman

I don't think it has a special name.... Just a series.

Why? Are you trying to sum it?

ye working out convergence dont know how.

Daniel Freedman

Original post by ibbi786

ye working out convergence dont know how.

What tests do you know that tell you about convergence?

OP

Original post by Daniel Freedman

What tests do you know that tell you about convergence?

not to sure was trying ratio test but got lost

Daniel Freedman

Original post by ibbi786

not to sure was trying ratio test but got lost

The ratio test states that, if you have

\displaystyle \sum_{n=1}^{\infty} a_n

, then:

- If

\displaystyle \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1

, the sum converges

- If

\displaystyle \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| > 1

, the sum diverges

- If

\displaystyle \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = 1

, then it doesn't tell us anything.

Have you tried calculating

\displaystyle \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|

? Where did you get stuck?

OP

Original post by Daniel Freedman

The ratio test states that, if you have

\displaystyle \sum_{n=1}^{\infty} a_n

, then:

- If

\displaystyle \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1

, the sum converges

- If

\displaystyle \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| > 1

, the sum diverges

- If

\displaystyle \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = 1

, then it doesn't tell us anything.

Have you tried calculating

\displaystyle \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|

? Where did you get stuck?

ive found out it converges but dont know how to calculate what it converges to...

Daniel Freedman

Original post by ibbi786

ive found out it converges but dont know how to calculate what it converges to...

Ok. I'll show you how to do it, but I'll spoiler sections appropriately to give you the opportunity to benefit from thinking about it. Please do think about it.

There's a nice trick to summing series like these that is useful to learn. The idea is to start off considering

(1-x)^{-1} = 1 + x + x^2 + x^3 + ... \ \ \ (*)

and then fiddle with it, to make the RHS look like our series.

(you can see this is true, as RHS is an infinite geometric series with common ratio x)

Now, if we substitute x = 1/2 into both sides, we get something that looks promising:

(1-\frac{1}{2})^{-1} = 1 + \frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3} + ...

It has the increasing powers of 1/2 that we need, but it doesn't have the right numbers on top (i.e. the 1, 2, 3 etc in our original series). Can you think of something we can do to (*), to get these numbers to appear?

Spoiler

[Note,]

(edited 13 years ago)

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