Convergence and Divergence

Unit 11.1 of Math 152

What is a series?

• when a funtion that is being added to infinity, not the terms to infinity!

Convergence

• Partial sums

  • if for example we have the series ${\sum }_{n=1}^{\infty }2n$ and we want to know if it converges we can check the sequence (the 2n part) and its partial sums

    • the sequence {sn} is convergent and ${\lim }_{n\to \infty }{s}_n=s$ exists as a real number, then the series is convergent
      • however since it grows forever it diverges

  • NOTE: this is different from the Divergence test!!!!

• Divergence test

  • to know if a sum is divergent we can use the divergent test

    • Definition:

      • if we want to know if ${\sum }_{n=1}^{\infty }{a}_n$ we can take the limit of $a_n$ to infinity ${\lim }_{n\to \infty }{a}_n$ and if the limit =/= 0 its divergent but if its 0 it it might be convergent

        • THIS IS DIFFERENT FROM CONVERGENCE PARTIAL SUMS!!!