Series - Continued

Series - Continued

The next couple of days are all about getting comfortable with some common series and how to work with the notations.

Let's Start Simple

• Let's start by taking a series in Sigma notation and expand it out.

• Be mindful of where the series starts and where it ends. Remember that it doesn't have to start at 0.

• In the series on the right, notice that the index of summation can be any variable you choose.

Multiple Choice

Expand $\sum_{n=2}^52^n$  

•  $2^2+3^2+4^2+5^2$  

•  $2^2+2^3+2^4+2^5$  

•  $2^2+2^5$   

•  $2^2+5^2$  

Let's go the other way

• We can take a series in expanded form and write it in sigma notation.

• The key is recognizing which kind of pattern is occurring.

• You get to decide your index of summation and where it begins.

• Let's try one out.

Multiple Select

The following sum: $\sqrt{2}+\sqrt{3}+\sqrt{4}+\sqrt{5}$ can be written in which of the following sigma notations? (select all that apply)

•  $\sum_{n=2}^5\sqrt{n}$  

•  $\sum_{n=3}^6\sqrt{n-1}$  

•  $\sum_{n=1}^{\infty}\sqrt{n}$  

•  $\sum_{n=1}^4\sqrt{n+1}$  

The Difference Between $s_n$  and  $a_n$  

•  $a_n$  is the expression that determines the value in a sequence.

•  $s_n$  is the expression that determines the partial sum of a sequence.

•  $s_n=\sum_{i=1}^na_i=a_1+a_2+a_3+...a_{\left(n-1\right)}+a_n$  

• This distinction is important when it comes to determining if a series converges or diverges.

Poll

 $a_n=\frac{2n+4}{3n+5}$ Does this SEQUENCE converge?

• Yes, the sequence converges

• No, the sequence does not converge

Poll

 $a_n=\frac{2n+4}{3n+5}$ Does this SERIES converge?

• Yes, the series converges

• No, the series does not converge

Divergence Test Broken Down

• Remember that $a_n$  is the expression that determines the value in the sequence. 

• If that value does not converge to zero it turns out that means the series diverges.

• Subsequently, just because  $a_n$  converges to zero, that does not mean its series converges. 

The Harmonic Series

• When we started learning about integration, one technique was for a family of functions that are written as  $\frac{1}{x^n}$  

•  $\frac{1}{x}$  was a special case of integration and is also a special case when it comes to series. 

• Although it's limit approaches zero as an expression, the limit of the series approaches infinity.

• There are multiple tests that can be used to determine if a series diverges and the harmonic series uses the integral test to show it's divergent.

Telescoping Sum

• These are extremely useful for repeating decimals.

• A telescoping series is a series where each term $u_k$ can be written as  $u_k=t_k-t_{\left(k+1\right)}$ for some series  $t_k$  
  

• An understanding of partial fraction decomposition tends to help break this down.