Series

Unit X.x of Math 152

Series

• Suppose $a_n$ is a sequence of numbers. An expression of the form

  • $a_1+a_2+a_3\cdots +a_n+\dots$

    • is called an infinite series and it is denoted by

      • ${\sum }^{\infty }$

• Definition

  • Given the series

    • ${\sum }_{i=1}^{\infty }{a}_i=a_1+a_2+\dots$ let s_n denote its $n^{th}$ partial sum
      • if the sequence {sn} is convergent and ${\lim }_{n\to \infty }{s}_n=s$ exists as a 0real number, then the series is convergent

        • s is the sum of the series

• Infinite series and geometric

  • infinite $1+\frac{1}{2}+\frac{1}{4}+\dots$ has the partial sums $s_1=1,s_2=1.5,s_3=1.75\dots$

    • and in general it turns out that $s_n=2-\frac{1}{2^{n-1}}$

  • Geometric. show that the geometric series ${\sum }_{i=1}^{\infty }a{r}^{i-1}=a+ar+a{r}^2+\dots$is convergent if $|r|<1$and its sum is

    • ${\sum }^{{\infty }_{i=1}a{r}^{i-1}}=\frac{a}{1-r},|r|<1$
      • if |r| is $\ge$ the series is divergent a=1
        • ${\sum }_{i=1}^n{r}^{i-1}=r{\sum }_{i=1}^n{r}^{i-1}+1-r^n$
          • $(1-r)s_n=1-r^n$ (assuming that r=/=1)
            • ⇒ $s_n=\frac{1-r^n}{1-r}$ ${\lim }_{n\to \infty }{s}_n=\frac{1}{1-r}$
      • if a is a coefficient we can rewrite it as
        • $a\sum$

• Examples.

  • Ex 1. Determine whether that the given series converges or diverges

    • a. ${\sum }_{n=1}^{\infty }(\frac{e}{10}{)}^n$

      • ⇒ ${\sum }_{n=1}^{\infty }\frac{e}{10}{\left(\frac{e}{10}\right)}^{n-1}$ ⇒

  • Ex 2. write 0.55555555… as a rational number

    • 0.55555…= $\frac{5}{10}+\frac{5}{100}+\frac{5}{1000}+\dots$
      • ⇒ ${\sum }_{1=1}^{\infty }5{\left(\frac{1}{10}\right)}^i⟹{\sum }_{i=1}^{\infty }\frac{5}{10}{\left(\frac{1}{10}\right)}^{i-1}$
        • $r=\frac{1}{10},a=\frac{5}{10}$
          • ⇒ $\frac{5}{9}$

  • Ex 3. show that the series ${\sum }_{n=1}^{\infty }\frac{1}{(n+1))(n+2)}$

    • ⇒ $\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\dots$

      • This is ugly and doesnt have any nice patterns so we can do partial fraction decomposition

        • ⇒ $\frac{1}{(n+1)(n+2)}=\frac{1}{n+1}-\frac{1}{n+2}$
          • $S_k={\sum }_{n=1}^k\frac{1}{(n+1)(n+2)}={\sum }_{n=1}^k\frac{1}{n+1}-\frac{1}{n+2}=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\dots -\frac{1}{k+2}$
            • = $\frac{1}{2}-\frac{1}{k+2}$
              • ⇒ ${\lim }_{k\to \infty }{S}_k=\frac{1}{2}$

  • Ex 4. Show that the harmonic series is divergent

    • ${\sum }_{n=1}^{\infty }\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\dots$
      • $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}$
      • we can group it so that
        • $a_{2^n+1}+\dots a_{2^{n+1}}\ge \frac{1}{2^{n+1}}{2}^n=\frac{1}{2}$
          • $S_{2^{n+1}}\ge \frac{1}{2}(n+1)⟹diverges$