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Series - survival kit

Definition of series

 

Given a sequence \(u_n\), you can play with it and create a new sequence \(S_n\) defined by

 \(S_1=u_1\)

\(S_2=u_1+u_2\)

\(S_3=u_1+u_2+u_3\)

More generally \(S_n=u_1+u_2+\cdots+u_n=\sum_{k=1}^{n}u_k\)

 \(S_n\) is called the sequence of partial sums.

It is a sequence and any results about sequences can be used for \(S_n\).

 

The series \(\sum u_n\) or \(\sum_{k=0}^{\infty} u_n\) is the limit of \(S_n\) when \(n\) goes to \(\infty\).

Basically a series is a real number, the value of the limit when \(S_n\) is convergent.

 

We say the the series is convergent when the sequence \(S_n\) is convergent, has a finite limit. In this case the series is equal to the value of the limit.

The series is divergent when the limit of \(S_n\) is infinite or doesn't exist.
 

Classic series

  • Geometric Series

 A geometric series with ratio \(r\) is  \( \sum_{k=0}^{\infty}r^k\).

We can prove that its sequence of partial sum \(S_n=\sum_{k=0}^{\infty} r^k=\frac{1-r^{n+1}}{1-r}\) for \(r\neq 1\) and \(S_n=n+1\) for \(r=1\).

Multiply \(S_n\) by \((1-r)\), after simplifications, you will find that \(S_n(1-r)=1-r^{n+1}\)

By taking the limit as \(n\) goes to \(\infty\), we can conclude that

 

 Theorem: The geometric series of ratio \(r\) is convergent for \(r\in(-1,1)\) and the series

$$\sum_{k=0}^{\infty}r^k=\frac{1}{1-r}.$$

The geometric series is divergent for \(r\notin(-1,1)\).

 

  • p-series

A p-series is a series in the form $$\sum_{n=1}^{\infty}\frac{1}{n^p}$$

 

 

 P-test: the p-series \(\sum_{n=1}^{\infty} \frac{1}{n^p}\) is convergent for \(p>1\)

The p-series is divergent for \(p\leqslant 1\).

 

Convergence criteria

 

Theorem: If a series \(\sum u_n\) is convergent then \(\lim_{n\rightarrow\infty}u_n=0\)

Remark: The converse is FALSE. The series \(\sum\frac{1}{n}\) is divergent and \(\lim_{n\rightarrow\infty}\frac{1}{n}=0\).

The contrapositive is TRUE and is probably the most used form of the theorem: if \(\lim_{n\rightarrow \infty}u_n\neq 0\) then the series is divergent.

Example: \(\lim_{n\rightarrow\infty}\frac{n^2+\sin n}{3n^2+\sqrt{n}}=\frac{1}{3}\neq 0\) therefore the series $$\sum_{n=1}^{\infty}\frac{n^2+\sin n}{3n^2+\sqrt{n}}$$ is divergent. 

 

Series with nonnegative terms

Theorem (Comparison test, Inequality version) Let \(\sum a_n\) and \(\sum b_n\) series with non negative terms such that for all the index greater than some \(N\), \(0\leqslant a_n\leqslant b_n\).

  • If \(\sum b_n\) is convergent, then \(\sum a_n\) is convergent.
  • If \(\sum a_n\) is divergent, then \(\sum b_n\) is divergent.

 

Theorem (Comparison test, limit version)  Let \(\sum a_n\) and \(\sum b_n\) series with positive terms such that \(\lim_{n\rightarrow\infty} \frac{a_n}{b_n}=c\neq 0\), then either \(\sum a_n\) and \(\sum b_n\) are both convergent or \(\sum a_n\) and \(\sum b_n\) are both divergent.

 

Theorem (Ratio Test) Let \(\sum a_n\) and \(\sum b_n\) series with positive terms such that \(\frac{a_{n+1}}{a_n}\leqslant \frac{b_{n+1}}{b_n}\) for any \(n\),

then

  • If \(\sum b_n\) is convergent, then \(\sum a_n\) is convergent.

  • If \(\sum a_n\) is divergent, then \(\sum b_n\) is divergent.

  •  

Corollary (Comparison to a geometric series) If for any \(n\), \(\frac{u_{n+1}}{u_n}<r<1\), then \(\sum u_n\) is convergent.

Corollary: If \(\lim_{n\rightarrow \infty}\frac{u_{n+1}}{u_n}=L\)

  •  if \(L>1\), then the series with general term \(u_n\) is divergent,

  •  if \( L<1\), the series with general terms \(u_n\) is convergent.

  • if \(L=1\), we cannot conclude.

Theorem (comparison with an integral)

Given a positive function \(f\) that is non increasing on a interval \([1,\infty)\).

Then the series \(\sum f(n)\) and \(\int_1^{\infty} f(t)d t\) are both convergent or they are both divergent.

 

Absolute convergence and conditional convergence

Definition Given a series \(\sum u_n\), \(\sum u_n\) is absolutely convergent if \(\sum |u_n|\) is convergent.

Theorem  If \(\sum u_n\) is absolutely convergent, then \(\sum u_n\) is convergent.

Definition If \(\sum u_n\) is convergent and not absolutely convergent, then \(\sum u_n\) is conditionally convergent.

Theorem (Alternating series) A series \(\sum u_n\) uch that its terms alternate between positive and negative, and such that \(|u_n|\) is decreasing toward 0 is convergent.