Question

Explain the convergence of the sequence an=1n graphically.
ALL STUDENTS
1
K2

2.
Show that series n=1∞-1n54n is convergent and find its limit.
1
K3

3.
Solve: x=1991x(x+1)
1
K3

4
Show that the series n=1∞(2n)!(n!)2 is not convergent.
1
K3

5.
Explain that the series n=1∞n2(2)n is convergent.
1
K3

6.
Show that the power series 1-12x-2+14(x-2)2-18(x-2)3+… converges to 2x for 0<x<4.
1
K3

7.
Determine the interval and radius of convergence for the power series n=0∞n3n+1xn.
1
K4

8.
Determine the interval and radius of convergence for the power series n=0∞10nn!(x-1)n.
1
K4

9.
Find the Taylor series generated by fx=1/x at x=2.
1
K3

10.
Find the Fourier sine and cosine series of the function fx=x in the interval 0<x<2.​

Answer 1

Answer:

This implies that

x2+2ax=4x−4a−13

or

x2+2ax−4x+4a+13=0

or

x2+(2a−4)x+(4a+13)=0

Since the equation has just one solution instead of the usual two distinct solutions, then the two solutions must be same i.e. discriminant = 0.

Hence we get that

(2a−4)2=4⋅1⋅(4a+13)

or

4a2−16a+16=16a+52

or

4a2−32a−36=0

or

a2−8a−9=0

or

(a−9)(a+1)=0

So the values of a are −1 and 9.