There was this question in my analysis exam today. I have a feeling it should be easy but no one I've asked knew how to do it.(adsbygoogle = window.adsbygoogle || []).push({});

We associate to the sequence [itex]\{a_n\}[/itex] the sequence defined by

[tex]b_n=\frac{a_1+a_2+...+a_n}{n}[/tex]

Show that if [itex]\{a_n\}[/itex] converges towards a, then [itex]\{b_n\}[/itex] converges towards a.

I realised that

[tex]b_n=\frac{\sum_{n=1}^{\infty} a_n}{n}[/tex]

or even

[tex]b_n=\frac{\sum_{k=1}^{n} a_k}{\sum_{k=1}^{n} 1}[/tex]

but all my attemps involving epsilon-delta, convergence tests, Cauchy convergence "caracterisation", etc. failed. Please tell me how to do this. Thanks a lot.

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Question in my test today

Loading...

Similar Threads for Question test today | Date |
---|---|

Divergence Test for an Infinite Series (General question) | Mar 18, 2014 |

Question about the Integral Test | Apr 13, 2013 |

Question about one part of the Ratio Test proof | Mar 15, 2012 |

Quick Question on Limit Comparison Test | Mar 1, 2010 |

Simple question about Integral Test | May 15, 2008 |

**Physics Forums - The Fusion of Science and Community**