54. If sz is the k th partial sum of the harmonic series, prove that In (k + 1) < S <1+ln k. Hint: 1 1 <-s- if 0

Q: 41. What are the first three terms of the Maclaurin Series for √x³? A. 0 B. C. ²x+3x2² D. 1/x+x² E.…

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Q: Let a, b e R such that 2a + b = 5 (1) If the series a b n n +2 n=3 is convergent then %3D (ii) If a,…

A: Given a,b∈R such that 2a+b=5 (i) Given the series ∑n=3∞an+bn+2 is convergent The we have to find the…

Q: 6. Find the power series and determine the interval of convergence of 3x f(x): x2+x-2

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Q: 3) Use the fact the Er =;- if -1<x<1 to find series representations for each of 1-x the following…

A: Solve the following

Q: Let x0, x1,x2,... be the sequence such that xo = 1 and for n ≥ 0, Xn+1 = ln(en — Xn) (as usual, the…

A: Let x0, x1, x2, ...... be the sequence such that x0=1. For n≥0, xn+1=lnexn-xn. We need to show that…

Q: Find the largest interval (in x) where the series summation from n = 1 to infinite of xe-nx…

A: Here we use D'Almbert ratio test.

Q: 4.- Suppose we have a number with infinite decimals of the form 0.d₁d₂d3d4... (where each d₂ = 0, 2,…

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Q: 7. Find a power series representation for the function and determine the interval of convergence. X…

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Q: ·(-1)* x* 5) Identify the function representation of the series E) * 4* and give its interval k = 0…

A: The given series∑k=0∞(-1)kxk4k

Q: 5. Use the Integral test to determine whether the series k converges or diverges Ak*+1 You do not…

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Q: 2. Prove that the p-series converges for p > 1.

A: