Mathematical Methods in the Physical Sciences/Chapter 1

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Sub Sections:

• /Section 1
• /Section 2
• /Section 3
• /Section 4
• /Section 5
• /Section 6
• /Section 7
• /Section 8
• /Section 9
• /Section 10
• /Section 11
• /Section 12
• /Section 13
• /Section 14
• /Section 15
• /Section 16

Overview

This is all about series!

Notes

• Infinite series converge if their sum exists, and diverge if their sum does not exist.
• Absolutely convergent series are series which converge even if all the terms were positive.
• Conditionally convergent series are series which converge, but if all the terms were positive it would diverge.
• The preliminary test tests if <math>\lim_{n\to=infty}a_n = 0</math>. If not, then the series is divergent. If so, then we need to test further.
• The comparison test compares the series to another known series.
  • If each term beyond some finite number is less than or equal to the corresponding terms of a convergent series, then the series converges.
  • If each term beyond some finite number is greater than or equal to the corresponding terms of a divergent series, then the series diverges.
  • This is the most useful test if you already know a ton of series.
• The integral test tests if the integral up to infinity for a series is finite. Beyond some finite number of terms, each succeeding term must be less than the previous term and greater than 0.
  • If the integral is finite, the series converges.
  • If the integral is infinite, the series diverges.
• The ratio test compares the limit of the ratio of succeeding terms.
  • If the ratio is less than 1, the series converges.
  • If the ratio is 1, then more tests are needed.
  • If the ratio is greater than 1, the series diverges.
• The special comparison test tests the limit of the series and another known series.
  • The series must consist of positive terms.
  • If the limit of the ratio with a convergent series is finite, then the series is convergent.
  • If the limit of the ratio with a divergent series is greater than 0 or infinite, the series is divergent.
• The alternating series test simply tests the limit of the absolute value of the terms. If it is 0, then the alternating series converges.
• The geometric series is <math>\sum_{n=1}^\infty ar^n</math>.
  • Its partial sum is <math>S_n = {a(1-r^n) \over 1-r}</math>
  • It converges to <math>S = {a \over 1 -r}</math> if <math>|r| < 1</math>, diverges otherwise.
• The harmonic series is <math>\sum_{n=1}^\infty {1 \over n}</math>.
  • Although it does not converge, it's alternating series does since <math>\lim_{n\to\infty}{1 \over n} = 0</math>.
• Absolutely convergent series are series where if each term were positive, it would still converge.
• Conditionally convergent series are series that converge but if all the terms were positive would diverge.

Series Algebra

• Convergence or divergence is not affecting by multiplying or changing a finite number of terms.
• Two convergent series may be added or subtracted term by term. The result is convergent, and the sum of the result is the sum of the sums of the two series.
• Absolutely convergent series may be reordered without affecting the convergence or the sum.
• Conditionally convergent series may not be reordered without affecting convergence or the sum.

Power Series

• The power series is <math>\sum_{n=0}^\infty a_nx^n</math> or <math>\sum_{n=0}^\infty a_n(x-a)^n</math>
• Power series may converge or diverge depending on the coefficients and the value of x.
• Power series may be differentiated or integrated term by term. The result converges to the integral or derivative of the corresponding function.

(TODO)