Real Analysis Help. Series Convergence

**megamet2000** · November 20th 2008, 01:17 PM

i would love if someone could help me with 2 problems im having trouble with.

1) Let $a_1, a_2, a_3$ , ... be a decreasing sequence of positive numbers. Show that $a_1+a_2+a_3+$ ... converges if and only if $a_1+2a_2+4a_4+8a_8+$ ... converges. i saw a similar one on here a couple days ago but this is slightly different

2) Show that a power series $\sum\limits_{n = 1}^\infty {c_nx^n }\$ has the same radius of convergence as $\sum\limits_{n = 1}^\infty {c_{n+m}x^n }\$ , for any positive integer m

thanks

**Plato** · November 20th 2008, 02:29 PM

Here is the problem: $a_n > a_{n + 1} > 0\,,\,\sum\limits_{n = 1}^\infty {a_n } \,\& \,\sum\limits_{n = 0}^\infty {2^n a_{2^n } }$ .

Using that $a_n$ is decreasing, then following shows it in one direction.
$\begin{gathered} a_1 + \underbrace {a_2 + a_3 }_{} + \underbrace {a_4 + a_5 + a_6 + a_7 }_{} + \underbrace {a_8 + a_9 + a_{10} + a_{11} + a_{12} + a_{13} + a_{14} + a_{15} }_{} + \cdots \hfill \\ \leqslant a_1 + 2a_2 + 4a_4 + 8a_8 \cdots \hfill \\ \end{gathered}$

Now note that:
$\begin{gathered} a_2 + \underbrace {a_3 + a_4 }_{} + \underbrace {a_5 + a_6 + a_7 + a_8 }_{} + \underbrace {a_9 + a_{10} + a_{11} + a_{12} + a_{13} + a_{14} + a_{15} + a_{16} }_{} + \cdots \hfill \\ \geqslant a_2 + 2a_4 + 4a_8 + 8a_{16}\cdots \hfill \\ \end{gathered}$
Also note that if $\sum\limits_{n = 1}^\infty {a_n }$ converges that $ \sum\limits_{n = 2}^\infty {2a_n }$ converges.

Can you finish?

**megamet2000** · November 20th 2008, 04:47 PM

Originally Posted by Plato

Here is the problem: $a_n > a_{n + 1} > 0\,,\,\sum\limits_{n = 1}^\infty {a_n } \,\& \,\sum\limits_{n = 0}^\infty {2^n a_{2^n } }$ .

Using that $a_n$ is decreasing, then following shows it in one direction.
$\begin{gathered} a_1 + \underbrace {a_2 + a_3 }_{} + \underbrace {a_4 + a_5 + a_6 + a_7 }_{} + \underbrace {a_8 + a_9 + a_{10} + a_{11} + a_{12} + a_{13} + a_{14} + a_{15} }_{} + \cdots \hfill \\ \leqslant a_1 + 2a_2 + 4a_4 + 8a_8 \cdots \hfill \\ \end{gathered}$

Now note that:
$\begin{gathered} a_2 + \underbrace {a_3 + a_4 }_{} + \underbrace {a_5 + a_6 + a_7 + a_8 }_{} + \underbrace {a_9 + a_{10} + a_{11} + a_{12} + a_{13} + a_{14} + a_{15} + a_{16} }_{} + \cdots \hfill \\ \geqslant a_2 + 2a_4 + 4a_8 + 8a_{16}\cdots \hfill \\ \end{gathered}$
Also note that if $\sum\limits_{n = 1}^\infty {a_n }$ converges that $ \sum\limits_{n = 2}^\infty {2a_n }$ converges.

Can you finish?

thanks for the help i appreciate it. proofs have never been my strong point would u mind helping me finish this?

also can anyone help me with the second question?

**Mathstud28** · November 20th 2008, 05:25 PM

Originally Posted by megamet2000

2) Show that a power series $\sum\limits_{n = 1}^\infty {c_nx^n }\$ has the same radius of convergence as $\sum\limits_{n = 1}^\infty {c_{n+m}x^n }\$ , for any positive integer m

thanks

http://www.mathhelpforum.com/math-he...er-series.html

**megamet2000** · November 20th 2008, 06:05 PM

awesome thank you very much

**megamet2000** · November 23rd 2008, 10:23 AM

does anyone have any tips on how to finish number 1?

**Mathstud28** · November 23rd 2008, 10:27 AM

Originally Posted by megamet2000

does anyone have any tips on how to finish number 1?

What are you having problems with? Plato practicially gave you the solution.

**megamet2000** · November 23rd 2008, 11:36 AM

i dont understand the purpose of the braces. i get how $a_n<2^na_{2^n}$ but after he says "now note that" i dont get how that sequence is now greater than the last sequence

**Plato** · November 23rd 2008, 12:27 PM

The first inequality tell us that $\sum\limits_{k = 1}^\infty {a_k } \leqslant \sum\limits_{k = 0}^\infty {2^k a_{2^k } }$ .
So if $\sum\limits_{k = 0}^\infty {2^k a_{2^k } }$ converges then $ \sum\limits_{k = 1}^\infty {a_k }$ converges.

Likewise, the second inequality tells us $\sum\limits_{k = 1}^\infty {2^{k - 1} a_{2^k } } \leqslant \sum\limits_{k = 2}^\infty {a_k }$ .
By comparison both converge.

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