OSU MTH265 Series – Consider the following series

 

First Name          Last Name           OSU Student ID #

Instructions: Do all of your work on this paper. Scan the completed problems and create one PDF of your work. Your PDF needs to be at least the same number of pages as the original docu- ment. Upload the PDF to this assignment in GRADESCOPE by the due date. (The Gradescope assignment can be accessed through Canvas.)

(1)          Consider the following series.

 ∞           n+1

 S =         (−1)        n

3n

n=1

(a)          Show that the series converges.

(b)          Estimate the error made when the partial sum S10 is used to estimate the series S.

(c)           What is the smallest value of N we could use to estimate the series to within 0.001 ?

 (2)         Consider the following series. For the work that follows, the root or ratio test will likely be convenient!

∞            n

 x

√n

n=1

(a)          Determine the value(s) of x (if any) for which the series converges absolutely.

(b)          Determine the value(s) of x (if any) for which the series converges conditionally.

(c)           Determine the value(s) of x (if any) for which the series diverges.

 (3)         The terms of a series can also be defined recursively. Suppose

1              n + 2

a1 = 4 and an+1 = 3n + 1 an for n ≥ 1

Use an appropriate test to determine whether or not the series converges.

(4)          Generalize the result of the previous exercise. Suppose a1 > 0 and an+1 = f (n)an for some function f . Give a condition on f (n) which would guarantee that such a series converges absolutely.

 (5)         Use either the root or the ratio test on the following series. State your conclusion.

 2+1 • 3 • 4

 3+2 • 4 • 5

 4+3 • 5 • 6

 54 • 6 • 7

 + . . .

 (6)         For the power series below, determine the radius of convergence and interval of convergence. For the interval, be sure to check the endpoints!

∞

n(x − 2)

n=0

 (7)         Use algebraic manipulation of power series to find the first three non-zero terms of the Taylor series for

ex ln(1 + x)

At most partial credit will be awarded for taking derivatives to find this instead of multi- plying the series. The Taylor series for ex and ln(1 + x) are on our posted list of common Taylor series.