Question
find the interval of convergence for the series
my question is #12 part b in the picture below it is;
there summation from n=1 to infinity of (2n)/(3(n!)) *x^n
MATH 224 J Nichos F15 SAMPLE FINAL EXAM NAME Do you want your course grade posted? Your secret code number is DIRECTIONS: Please, show al work! Use of a calculator is permitted (16) 1 Find the first derivative of each of the following (16) 2 Evaluate each of the limits - show your work f(x)-e-x+sec(In x ) b y=(sin x)'x.scf(x)-een +arcsin(e" ) a 1-cosx lim x1nx lim d lim (1 +-)s. c. (6) 3. (21) 4 Find the equation of the tangent line to y=lne'+e")at(0,1 2). Evaluate each of the following if it exists x2 In x dx (24) 5. Evaluate each of the following it it exists dx 3 dx x'(x+4) (6) 6 Define each of the following The series a- converges to S a b The sequence fa, converges to L (16) 7. Determine if the series is convergent or divergent EXPLAIN! If it is convergent, find its sum. (6) 8 ne is convergent or divergent Use the Integral test to determine whether the senes Determine f each of the following senes is absolutely convergent, conditionally convergent, or devergent EXPLAIN WHY 05) 9 Sinn (2n) In n (12) 10 Find the radius of convergence for each of the following 7-13.. (6n + (6n) 1- x -5 I n1 (6) 11. Use an infinite series to approxsimate the integralhx with an error less than 000001 sin x2 dx correct to within an error of 00001 (6) 11 Evaluate J0 (18) 12 Find the interval of convergence for each of the following n! 3(n)
Explanation / Answer
summation from n=1 to infinity of (2n)/(3(n!)) *x^n
an= (2n)/(3(n!)) *xn
an+1= (2(n+1))/(3(n+1)!) *xn+1
ratio test fir convergence :
limn->infinity |an+1/an| <1
limn->infinity |((2(n+1))/(3(n+1)!) *xn+1) /( (2n)/(3(n!)) *xn)| <1
limn->infinity |((2((n+1)/n)/(n+1)) *x 1) | <1
limn->infinity |((2(1 +1/n)/(n+1)) x) | <1
|((2(1 +0)/(infinity+1)) x) | <1
|2x/infinity|<1
0<1
interval of convergence =(-,)
