Intro to Analysis

[mwande03]

If

[FONT=MathJax_Math]a[FONT=MathJax_Math]n[/FONT][/FONT]​

is a sequence such that

[FONT=MathJax_Size1]∑[FONT=MathJax_Main]∞[/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Math]b[/FONT][FONT=MathJax_Math]n[/FONT][/FONT]​

converges whenever

[FONT=MathJax_Math]b[FONT=MathJax_Math]n[/FONT][/FONT]​

is a subsequence of

[FONT=MathJax_Math]a[FONT=MathJax_Math]n[/FONT][/FONT]​

, then

[FONT=MathJax_Size1]∑[FONT=MathJax_Main]∞[/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Math]a[/FONT][FONT=MathJax_Math]n[/FONT][/FONT]​

is absolutely convergent. (A series such as

[FONT=MathJax_Size1]∑[FONT=MathJax_Main]∞[/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Math]b[/FONT][FONT=MathJax_Math]n[/FONT][/FONT]​

is called a subseries of

[FONT=MathJax_Size1]∑[/FONT][FONT=MathJax_Main]∞[/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Math]a[/FONT][FONT=MathJax_Math]n[/FONT]​

.)

Any ideas?

Can I take c(n) subseq of a(n) such that every positive element of a(n) is an element of c(n). d(n) similarly picks up all the negative elements of a(n). Summation of c(n) minus summation d(n) is bounded by hypothesis, so a(n) is absolutely convergent. does that follow?

 

[daon2]

If \(\displaystyle \sum_{n=1}^{\infty} b_n\) converges whenever \(\displaystyle \{b_n\}\) is a subsequence of \(\displaystyle \{a_n\}\) then \(\displaystyle \sum_{n=1}^{\infty} |a_n|\) converges.

If only finitely many \(\displaystyle a_n\) are negative then let \(\displaystyle a_{k_1},...,a_{k_m}\) be the negative values and let \(\displaystyle S=\sum |a_{k_i}|\). Then let \(\displaystyle b_n=a_n\). We have \(\displaystyle \sum_{n=1}^{\infty} |a_n| = \sum_{n=1}^{\infty}b_n + 2S < \infty\)

You can show similarly for finitely many positive terms.

Now what is left to consider is infinitely many negative and positive terms. Consider the subsequences \(\displaystyle \{a_n;\,\, a_n \ge 0\}\) and \(\displaystyle \{a_n;\,\, a_n < 0\}\)