Q-1) Let f(x) = (-1)- n=0 2n+1 (2n+1)!* (i) Find all x for which this series...

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Q-1) Let f(x) = (-1)"- n=0 2n+1 (2n+1)!* (i) Find all x for which this series converges absolutely. (ii) Show that f(x) satisfies the differential equation y" + y = 0. (iii) A theorem on differential equations says that if g(x) is a solution of the above differential equation, then g(r)=Acos. + Bsin, where A and B are some constants. Show that f(x) = sinx. Show your work in detail. Correct answers without detailed explanation do not get any credit. Grading: 4+3+3=10 points. Answers: Q-1) Let f(x) = (-1)"- n=0 2n+1 (2n+1)!* (i) Find all x for which this series converges absolutely. (ii) Show that f(x) satisfies the differential equation y" + y = 0. (iii) A theorem on differential equations says that if g(x) is a solution of the above differential equation, then g(r)=Acos. + Bsin, where A and B are some constants. Show that f(x) = sinx. Show your work in detail. Correct answers without detailed explanation do not get any credit. Grading: 4+3+3=10 points. Answers: