MAT1236Exam 3Information

Date and Time5/31Wednesdays

SectionsSection 11

Total Points60 points

Sequences 5 tools you can use to find limits

1. If and , then .

2. The Limit Laws

3. The Squeeze Theorem

If and , then .

4. If , then .

5.

Standard Series

1. Geometric Series is .

In the case of convergent, .

2. Harmonic Series is divergent.

3. p-series is .

4. Alternating Harmonic Series is convergent.

Tests

The Test For Divergence

If or DNE, then is divergent.

The Integral Test

If is continuous, positive, decreasing on , and ,

then both , converge/diverge.

  • Note that in this case, for all .

The Comparison Test

Suppose , for all .

If is convergent, then is also convergent.

If is divergent, then is also divergent.

The Limit Comparison Test

Suppose .

If (is a finite no.) then both , converge/diverge.

The Alternating Series Test

If an alternating series or satisfy

(i) , for all

(ii)

then, or are convergent.

The Test For Divergence for Alternating Series Suppose =or

If , then . Thusis divergent.

The Ratio Test/ Root Test

or

Power Series

is the Radius Of Convergence of if is convergent for and divergent if .

End points of the Interval Of Convergence =

if I.O.C. =

if I.O.C. =

Representation of Functions as Power Series

If , then

and

  • We can determine the constant by plug in a particular value of .
  • The three series have the same radius of convergence.
  • The I.O.C. of the three series may be different at the end points.

Taylor Series Maclaurin Series

Power Series from Geometric Series

The series is convergent if and divergent if .

Other Concepts

Partial Fractions, Partial Sum Sequence, Telescoping Sum, Index Shifting of Summation

Important Notice

It is extremely important that you use the logical formats and show the necessary details. Answers alone or without proper support arguments will be given no points.

Points will be taken off for bad or incorrect notations e.g. writing integral notation without the differential “” or claiming is a p-series.

Practice Problems

(Disclaimer: This practice exam hasno direct relations with the real exam. You need to understand that the problems in the real exam may not resemble the homework problems or the problems in this practice exam. )

1. Determine whether the series converges or diverges:

We will use 4 different methods to find the convergence of the series.

1. Use Partial Sum Sequence2. Integral Test

3. Limit Comparison Test4. Comparison Test

Note that if the comparison between and is not obvious, you need to justify it. Sometimes, it is easier to show (which is equivalent to or ).

2. Determine whether the series is absolutely convergent, conditionally convergent, or divergent.

3. Find the radius of convergence and interval of convergence of the series

4. Find the Taylor series forcentered at by using the definition of Taylor series. (You need to provide sufficient details to support your conclusion.)

5. Given . Evaluate as an infinite series.

Answers

2. Divergent. 3. , I.O.C.= 4.

5.

Exam 3 PPFTNE (Note that this is not an exhausted list.)

1. State and prove the convergence of the geometric series.

Geometric Series
is .
In the case of convergent, .
Proof
If , and
If ,

If ,
If ,

2. State the Integral Test. With the help of a diagram, explain

(i) why the convergent of implies the convergent of .

(ii) why the convergent of implies the convergent of .

The Integral Test If is continuous, positive, decreasing on , and , then both , converge/diverge.
(i) Suppose is convergent. As illustrated in the diagram, can be interpreted as the area under on which is bigger than the sum of the area of the rectangles,. Since the only way that can diverge is to approach infinity, which is impossible. Thus, converges. Since finite number of terms will not change the convergence of a series, also converges. /
(ii) Suppose is convergent. As illustrated in the diagram, can be interpreted as the area under on which is smaller than the sum of the area of the rectangles,. Since the only way that can diverge is to approach infinity, which is impossible. Thus, converges. /

3. State the Squeeze Theorem for sequences.

If and , then .

4. Define the partial sum sequence of a series.

Given a series , for , we define its partial sum sequenceas .

5. Define the convergence of a series.

Let be the partial sum sequence of .
If is convergent and then is convergent and . Otherwise is divergent.

1