For each of the following series, determine whether the absolute convergence series test determines absolute convergence or fails. For the ¿th series, if the test is inconclusive then let Mi = 4, while if the test determines absolute convergence let Mi 1 : 2: ∞ Σ(−1)"+¹ sin(2n); n=1 Σ n=1 Σ ((−1)”. COS n² 3+2n4 3: (+ 4: 5 : n=1 ∞ n 2+5n3 ПП n² 2 5+2n3 пп n² Σ(+)+ n=1 ∞ n=1 COS 4 2 3+8n3 П ηπ n- (−1)+1 sin (+727) 5 + 2m³ 4 = 8. Then the value of cos(M₁) + cos(2M2) + cos(3M3) + sin(2M) + sin(M5) is -0.027 -0.621 -1.794 -1.132 -1.498 -4.355 -2.000 2.716

Find the interval of convergence of the power series. answer is a finite set of values, enter your answers as Σ(-1)"+¹(n+3)x² n=0

What is the radius of convergence R of the series. R=?

Q1. Test the following series for being convergent or divergent: (In n )"/ 2 b.) Ž 3n n 1 a.) 2 n=1 Q2.Use n-th term test to test the series: (-1)" Vn+ \n - Nn. n=1 1+e° +e2° + e +...+.. +. = 10 : 3c Q3 The value of (c) for which Q4. Choose the correct answer: a) The harmonic series is: i) Convergent. ii) Divergent. iii) Geometric. b) The limit of the n-th term of the series n tan( -) is: n=0 i) infinity ii) 1.0 iii) zero. c) The series E(-)" has: n=0 T i) no sum ii) sum=r iii) sum= (t/( T-e)).

Find the interval of convergence for each power series. If the interval of convergence is finite, be sure to analyze the convergence at the endpoints.

If a power series for some function y = g(x) has interval of convergence (−2, 2], is itpossible for the series for f′(x) to have interval of convergence [−2, 2]?Explain your answer.

Find the series' interval of convergence and, within its interval, the sum of the series as a function of ?x.

Find the radius and interval of convergence for the following series: (x - 2)" 10n a. n=0 Radius= Interval= Σ n=0 Radius= b. (-1)"x" n! Interval= C. Σ(n!) (x - 4)" n=0 Radius= Interval=

Find the interval of convergence of the power series -(x + 3)³n . n=1 n 27"

6. Express the power series: >(-3)"(x)"-1 n=0 As a function. Then find its radius and interval of convergence.