Name:

Directions: Show your work! Answers without justification will likely result in few points. Your written work also allows me the option of giving you partial credit in the event of an incorrect final answer (but good reasoning). Indicate clearly your answer to each problem (e.g., put a box around it). Good luck!

Problem 1 (25 pts). Determine whether the following sequences and series are convergent or divergent (give valid reasons!). For those which are convergent find the limit.

1. 

   We can rewrite this as

   

   Convergent, by exhibition of the limit!

2. 

   

   Convergent, as a monotone and bounded sequence converges to a limit.

3. 

   For this infinite series to converge, we must have that the terms tend to zero as : i.e.,

   

   This is clearly not the case, as the argument to cosine would have to approach multiples of , and an integer argument of n will not do so! Therefore, the series is divergent.

4. 

   This simplifies to

   

   and we can show that

   

   since is a continuous function of its continuous argument on this interval . Hence, the sequence is convergent with limit 0.

5. 

   This is divergent, by comparison with

   

   

   And since diverges, so does the given series.

   You could also proceed by the limit comparison test with , and then show that

   

   diverges by comparison with the harmonic series, for example.

Problem 2 (5 pts). Express as an infinite series.

As a check, evaluate

Problem 3 (10 pts). Analyze the integral

determining conditions on p for which it converges.

You should use the definition of the improper integral to do your analysis (that is, use limits).

First of all, we note that if then the integral is perfectly well behaved. Hence we consider in detail only the cases p>0.

1. p=1 is special case:

   

   which diverges, since the limit does not exist as of .

2. p<1:

   

   converges.

3. p>1:

   

   diverges, since the anti-derivative blows up as .

Thus, the answer is that for p<1 the integral is perfectly convergent; otherwise it diverges.

Problem 4 (10 pts). Give examples of series for which

1. 

2. 

3. 

4. 

Problem 5 (10 pts).

1. State in precise terms (i.e., using ) what it means to say that the sequence converges to the limit L.

   

2. From your definition (and not using the algebra of limits) prove that

   

   Given .

   Consider

   

   We now find where

   

   We take N to be the first positive integer greater than x. Then

   

   Q.E.D.

Problem 6 (10 pts). Decide whether the following integrals are convergent or divergent (give reasons!):

1. 

   

   The problem occurs at x=2, where the integral is improper. I should be written as the sum of two integrals,

   

   Neither one converges, as we can see by comparison, for example:

   

   which, upon u-substitution, looks like

   

   (and which fails to converge).

2. 

   This converges by comparison:

   

Problem 7 (5 pts). Find the values of p for which the series is convergent:

Obviously if we're divergent, as the terms will grow at least linearly. Hence consider p<0. We'll use the integral comparison test. Consider

Now f is clearly positive on the interval , but it needs to be eventually decreasing if we're to use this test. Thus we consider the derivative:

This must be negative for large values of x: that is, in the limit as , . Thus

Thus, only in the event that will this condition be satisfied. For , f is increasing, which means the that series diverges (since the limit of the terms cannot be zero). Thus, we can now restrict to .

We use a u-substitution to write this as

which converges if and only if p<-1. This then, is our answer: the series converges if and only if p<-1.

Problem 8 (15 pts). Consider the series

 

1. Demonstrate that the series converges, and use your calculator to estimate the limit.

   This converges by the alternating series test, as

   1. 

      

      and

      

   2. The calculator will give rather strange results, however, as this series converges slowly: I considered and , and got an average of .601481. The partial sum !

2. Plot the first 10 terms of the sequence and the first 10 partial sums of the series (1) on the same axes. Indicate clearly which graph is which.

     
   Figure 1: Problem 8: Terms in black, sums in blue.
   

3. How many terms m of the series (1) do we need to add in order to assure that the size of the error incurred by approximating the series by the partial sum is less than .0001?

   In order to assure that the error is less than .0001, we need to find m such that

   

   Hence we solve , or

   

   Thus, terms should work.

Problem 9 (10 pts). Consider the series

 

1. Use the sum of the first 20 terms to estimate the sum of the series.

   1.2008678419584369

2. Bound the error of this estimate (that is, find values a and b such that ``tightly'').

   

   

   Hence

   

Problem (Extra Credit: 5 pts). There are 2 million points in the plane, no three of which are colinear. Is it possible to draw a line through them so that exactly 1 million are on one side, and 1 million are on the other?

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