Using the bionomial series

One problem is extra credit. Duration: 50 min. Be neat, show your work to receive credit. 1 1/10 sin x (5 points) Using the power series, estimate dx with an error of less than 5 x 10-6. 1 + x Hint: divide sin x by 1 + x using long division to get the first 4 terms. L” 2 1- 31 + 2x (5 points) Using the bionomial series of f(x) = { 1 + 2x evaluate the limit lim x0 х 3 (-1)”(x – 2)” (5 points) Determine the interval of convergence of the series 8 WE n 3п n=1 5 (5 points) After finding the first 4 terms of the series of f(x) = e-* sec x, estimate f(0.1) correct to 2 decimal places. Hint: long division works in ex COS X 6 in powers of x – 3. Find the interval of convergence of the 1 (5 points) Find a series representation of f(x) = 3 + 2x resulting series. 4 1 -1 (5 points) Find the exact values of the series below using sin x, cos x, er, and tan χ. 1-x X ΧΟ ΧΟ 2n-1 (α) Σ (-1)”π2n+1 (b) 4η (2n + 1)! n=0 (-1)”12n (c) Σ (2n)! (d) Σ (-1)”” 3nn! n=0 (e) Σ (-1)” 2n +1′ n=0 3n+1 n=0 n=0 7 (5 points) Determine whether the series below is convergent or divergent. Write your decision right below each series, and rigorously explain your reasoning to receive credit. n5 1 12 n” 1 Ι) Σ Ι) Σ III) Σ ΙV) Σ V) Σ 2n 1+ on n! 5η n Inn ΧΟ Σ Σ n4 +n n=1 n=1 n=1 η=1 η=3
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