# Thread: Real Analysis Help. Series Convergence

1. ## Real Analysis Help. Series Convergence

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

2. 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?

3. 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?

4. 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

5. awesome thank you very much

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

7. 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.

8. 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

9. 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.