MATH1012 - Lecture 19

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MATH1012 - Lecture 18
Skeleton Slides

A Test for Divergence

• For a series ${\sum }_{n=1}^{\infty }{a}_n$ to converge to a value $L$, $\underset{n\to \infty }{\lim }{a}_n\ne 0$
  • (Double check the last part)
• Harmonic series: ${\sum }_{n=0}^{\infty }\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots$
  • Harmonic series are always divergent (see slides for proof)
• Geometric series: ${\sum }_{n=0}^{\infty }{r}^n=1+r+r^2+\cdots$
  • A geometric series converges to $\frac{1}{r-1}$ only if $|r|<1$
  • $s_N=\frac{1-r^{N+1}}{1-r}$
    • Assuming you start at the 0th term
    • Otherwise, minus $1(r^0)$ from $s_N$

Series Law

• If $S_a={\sum }_{n=1}^{\infty }{a}_n$ and $S_b={\sum }_{n=1}^{\infty }{b}_n$ are convergent series, then:
  • $S_{a\pm b}=S_a\pm S_b$
  • $S_{ka}=k{S}_a,k$ is a constant
  • $\text{Note: }{S}_a{S}_b\ne S_{ab}$ and $S_a/S_b\ne S_{a/b}$

If ${\sum }_{n=1}^{\infty }{a}_n$ or ${\sum }_{n=1}^{\infty }{b}_n$ diverges, so does ${\sum }_{n=1}^{\infty }(a_n\pm b_n)$
If ${\sum }_{n=1}^{\infty }{a}_n$ diverges, so does ${\sum }_{n=1}^{\infty }k{a}_n$, for $k\ne 0$

• As we can use our series laws if we know our series are convergent, it is important we have numerous ways of testing if a series is convergent
• Below and in other lectures, we will look at these methods including:
  • Integral Test
  • Comparison Test
  • Limit Comparison Test
  • Alternating Series Test
  • The Ratio Test

The Integral Test

• With a sequence $a_1,a_2,a_3,\cdots$ of positive values and a positive, decreasing continuous function $f$ such that $f(n)=a_n$:
  • Then ${\sum }_{n=1}^{\infty }{a}_n$ and ${\int }_1^{\infty }f(x)$ both converge or both diverge
  • Note: that the integral is improper so it may not exist (diverge)
    • Typo on slides
• If the integral is bounded than the series converges
  • Otherwise, the series diverges
  • See Improper Integrals

Keep in mind, the integral tells you nothing about the actual series

• For a series in the form ${\sum }_{n=1}^{\infty }\frac{1}{n^p}$, if $p>1$, the series is convergent
  • Otherwise, the series diverges

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MATH1012 - Lecture 20
Annotated Slides