1. (12) Give the first five terms of a sequence and find its limit. 2. (7) Calcu

Question

1. (12) Give the first five terms of a sequence and find its limit. 2. (7) Calculate the sum of a geometric series. 3. (12) Test a given series for convergence using the integral test. 4. (12) Calculate how many terms are required to ensure the sum of a given series to within a certain accuracy. 5. (32) Test various series to see if they converge or diverge. 6. (15) Use the ratio test to find the radius and interval of convergence of a given series. 7. (8) Give formula for the nth term of the power series of a given function and determine its radius of convergence. 8. (12) Find terms of a Maclaurin series of a product of functions. 9. (10) Expand a function using the binomial series and give its radius of convergence. 10. (30) Given a function: find the first few terms of its Taylor series about a given point, write a Taylor polynomial and use it to calculate an approximation, and give the formula for a Taylor remainder and use it to obtain a good bound at a given point.

Explanation / Answer

3)let function f(x)=xe-x^2

a)function is positive

b)function is continous

c)f(x)=xe-x^2

f '(x)=e-x^2 +xe-x^2(-2x)

f '(x)=e-x^2 (1+x(-2x))

f '(x)=e-x^2 (1-2x2)

increasing ==>e-x^2 (1-2x2)>0

==>(1-2x2)>0

==>2x2<1

==>x=(-1/sqrt2, 1/sqrt2)

upto x= 1/sqrt2 function is increasing

decreasing ==>e-x^2 (1-2x2)<0

==>(1-2x2)<0

==>2x2>1

==>x=( 1/sqrt2,infinity)

from x= 1/sqrt2 function is decreasing

so eventually function is decreasing

so integral test can be applied

limt->infinityintegral[0 to t]xe-x^2 dx

=limt->infinity[0 to t](-e-x^2)/2

=limt->infinity(-e-t^2)/2 -(-e-0^2)/2

=limt->infinity(-e-t^2)/2 -(-1)/2

=limt->infinity(1-e-t^2)/2

=(1-0)/2

=1/2

integral is convergent so series sum[m=1 to infinity] ml-m^2 is convergent

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