第一題

(1 point) Consider the series
∑n=1∞(4x)nn.∑n=1∞(4x)nn.
Find the interval of convergence of this power series by first using the ratio test to find its radius of convergence and then testing the series' behavior at the endpoints of the interval specified by the radius of convergence.
interval of convergence = 
(Enter your answer as an interval: thus, if the interval of convergence were −3<x≤5−3<x≤5, you would enter (-3,5]. Use Inf for any endpoint at infinity.)

 

 

第二題

1 point) If ∑Cn(x−4)n∑Cn(x−4)n converges at x=9x=9 and diverges at x=13x=13, what can you say about:

(a) the convergence at x=16x=16?

(b) the convergence at x=14x=14?

(c) the convergence at x=11x=11?

(d) the convergence at x=0x=0?

 

 

第三題

(1 point) Find all the values of x such that the given series would converge.
 
∑n=1∞xnln(n+9)∑n=1∞xnln⁡(n+9)
Answer: 

 

 

第四題

(1 point) Match each of the power series with its interval of convergence.

 1. ∑n=1∞(x−8)n8n∑n=1∞(x−8)n8n
 2. ∑n=1∞(6x)nn8∑n=1∞(6x)nn8
 3. ∑n=1∞n!(6x−8)n8n∑n=1∞n!(6x−8)n8n
 4. ∑n=1∞(x−8)n(n!)(8)n∑n=1∞(x−8)n(n!)(8)n


A. (0,16)(0,16)
B. [−16,16][−16,16]
C. (−∞,∞)(−∞,∞)
D. {8/6}

 

 

第五題

(1 point) A famous sequence fnfn, called the Fibonacci Sequence after Leonardo Fibonacci, who introduced it around A.D. 1200, is defined by the recursion formula
f1=f2=1,fn+2=fn+1+fn.f1=f2=1,fn+2=fn+1+fn.
Find the radius of convergence of
∑n=1∞fnxn.∑n=1∞fnxn.
Radius of convergence:  .

 

第六題

(1 point) Consider the power series

∑n=1∞2⋅4⋅6⋅⋯⋅(2n)1⋅3⋅5⋅⋯⋅(2n−1)xn.∑n=1∞2⋅4⋅6⋅⋯⋅(2n)1⋅3⋅5⋅⋯⋅(2n−1)xn.
Find the radius of convergence R.R. If it is infinite, type "infinity" or "inf".
Answer: R=R= 

 

What is the interval of convergence?
Answer (in interval notation): 

 

 

 

 

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