a) the domain of the function is the interval of convergence of power series. Let an=n!(n+1)!22n+1(−1)nx2n+1 and find the limn→∞∣anan+1∣
∣n!(n+1)!22n+1(−1)nx2n+1(n+1)!((n+1)+1)!22(n+1)+1(−1)n+1x2(n+1)+1∣=∣(n+1)!(n+2)!22n+3x2n+3×x2n+1n!(n+1)!22n+1∣
=∣(n+1)n!(n+1)(n+2)!22x2×n!(n+1)!∣
=∣4(n+1)(n+2)x2∣
→0 as n →∞
By ratio test, Since limn→∞∣anan+1∣=0≤1 ∀ x , The given series converges ,With this ,write write the interval which is exactly the domain of the function
(−∞,∞)
b)
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