Expand f(2) = in powers of (z – 12). Also clearly mention the domain in which this expansion is valid (i.e., find the region of convergence).
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- Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive.∑n=1∞enn!ρ=limn→∞|an+1an|= (Enter 'inf' for ∞.) ∑n=1∞enn! is:A. convergentB. divergentC. The Ratio Test is inconclusive(b) The series converges for every x in the half-open interval [−1, 1) but does not convergewhen x = 1. For a fixed x0 ∈ (−1, 1), explain how we can still use theWeierstrass M-Test to prove that f is continuous at x0.determine the radius and convergence of E n=1 5xn/3n2
- Find the Taylor's series expansion centered at x=0 for the function f(x)= x2/(1+3x).The fourth term of the Taylor series of f(x)=lnx at x=2 isFind the pointwise limit f(x) for {nxe-nx} for x ∈ (0, +inf)). Does the sequence converge uniformly for x ∈ (0, +inf))? If yes, what is the uniform norm of fn(x)-f(x) on (0, +inf)?