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Let fn be the n-th Fibonacci number of Example 3c in 8.1, k=n k=n An = Σκ=1+2+...+n, Bnk²= 1² +2²+...+n²; k=1 k=1 by definition fo Ao = Bo = 0. (a) Give a recursive definition of the numbers fn, An, Bn with n≥ 0 (b) Use mathematical induction and only part (a) to show that fn, An, Bn ≤5" for all n≥0 (c) Use the Absolute Convergence and Comparison Tests and only part (b) to show that the power series ∞ DO ∞ f(x) = fnx", A(x) = Σ Ana", B(x)=Bnx", n=0 n=0 n=0 converge if x < 1/6 (and thus determine smooth functions near x=0).

Let fn be the n-th Fibonacci number of Example 3c in 8.1, k=n k=n An = Σκ=1+2+...+n, Bnk²= 1² +2²+...+n²; k=1 k=1 by definition fo Ao = Bo = 0. (a) Give a recursive definition of the numbers fn, An, Bn with n≥ 0 (b) Use mathematical induction and only part (a) to show that fn, An, Bn ≤5" for all n≥0 (c) Use the Absolute Convergence and Comparison Tests and only part (b) to show that the power series ∞ DO ∞ f(x) = fnx", A(x) = Σ Ana", B(x)=Bnx", n=0 n=0 n=0 converge if x < 1/6 (and thus determine smooth functions near x=0).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let fn be the n-th Fibonacci number of Example 3c in 8.1, k=n k=n An = Σκ=1+2+...+n, Bnk²= 1² +2²+...+n²; k=1 k=1 by definition fo Ao = Bo = 0. (a) Give a recursive definition of the numbers fn, An, Bn with n ≥ 0 (b) Use mathematical induction and only part (a) to show that fn, An, Bn ≤5" for all n≥0 (c) Use the Absolute Convergence and Comparison Tests and only part (b) to show that the power series ∞ DO ∞ f(x) = fnx", A(x) = Σ Ana", B(x) = Σ Bnt", n=0 n=0 n=0 converge if r < 1/6 (and thus determine smooth functions near r=0).
Transcribed Image Text:Let fn be the n-th Fibonacci number of Example 3c in 8.1, k=n k=n An = Σκ=1+2+...+n, Bnk²= 1² +2²+...+n²; k=1 k=1 by definition fo Ao = Bo = 0. (a) Give a recursive definition of the numbers fn, An, Bn with n ≥ 0 (b) Use mathematical induction and only part (a) to show that fn, An, Bn ≤5" for all n≥0 (c) Use the Absolute Convergence and Comparison Tests and only part (b) to show that the power series ∞ DO ∞ f(x) = fnx", A(x) = Σ Ana", B(x) = Σ Bnt", n=0 n=0 n=0 converge if r < 1/6 (and thus determine smooth functions near r=0).
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