Difference between revisions of "Mathematical Methods in the Physical Sciences/Chapter 1/Section 4"
(Created page with "== Overview == == Things to Remember == * Infinite series are either convergent (have a sum) or divergent (don't have a sum). * Playing with the sums of divergent series lea...") |
(→Things to Remember) |
||
Line 9: | Line 9: | ||
* The '''alternating harmonic series''' is convergent, but can be made to have any sum by rearranging the terms! | * The '''alternating harmonic series''' is convergent, but can be made to have any sum by rearranging the terms! | ||
* The partial sum of any series is: | * The partial sum of any series is: | ||
− | *:<math>S_n | + | *:<math>S_n = a_1 + a_2 + a_3 + ... + a_n</math> |
* The sum of a series is: | * The sum of a series is: | ||
*:<math>S = \lim_{n \to \infty} S_n</math> | *:<math>S = \lim_{n \to \infty} S_n</math> | ||
Line 17: | Line 17: | ||
* For convergent series, the remainder tends to 0 as n approached infinity. | * For convergent series, the remainder tends to 0 as n approached infinity. | ||
* Divergent series have no sum and thus no remainder. | * Divergent series have no sum and thus no remainder. | ||
− | |||
== Problems == | == Problems == | ||
[[../Section 5|Next Section]] | [[../Section 5|Next Section]] |
Latest revision as of 18:59, 18 April 2012
Overview
Things to Remember
- Infinite series are either convergent (have a sum) or divergent (don't have a sum).
- Playing with the sums of divergent series leads to problems. Don't do it.
- Sometimes a series of positive terms is divergent, but the corresponding alternating system is not.
- The harmonic series is <math>1 + {1 \over 2} + {1 \over 3} + ... + {1 \over n} + ...</math> and is divergent.
- The alternating harmonic series is convergent, but can be made to have any sum by rearranging the terms!
- The partial sum of any series is:
- <math>S_n = a_1 + a_2 + a_3 + ... + a_n</math>
- The sum of a series is:
- <math>S = \lim_{n \to \infty} S_n</math>
- If the sum does not exist, the series is divergent. If it does, it is convergent.
- The remainder of a convergent series is:
- <math>R_n = S - S_N</math>
- For convergent series, the remainder tends to 0 as n approached infinity.
- Divergent series have no sum and thus no remainder.