Evaluate the series: Sum of 5/6^n from n = 0 to infinity.
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x dx divergent (p-test). 2. ∫ ∞. 1 e−x dx convergent (integrate) ... xe−x dx convergent (integrate by parts) ... If ∫ f is convergent and 0 ≤ g(x) ≤ f(x) for all x ...
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Part a
∞∑n=1(3n)4=∞∑n=134n4=81∞∑n=11n4=81π490∑n=1∞(3n)4=∑n=1∞34n4=81∑n=1∞1n4=81π490
Part b
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∞∑k=61(k−3)4=∞∑k=31((k+3)−3)4[Drop the index from k=6 to
k=3]=∞∑k=31k4=∞∑k=11k4−114−124=π490−1716
∞∑n=1(3n)4=∞∑n=134n4=81∞∑n=11n4=81π490∑n=1∞(3n)4=∑n=1∞34n4=81∑n=1∞1n4=81π490
Part b
☛
∞∑k=61(k−3)4=∞∑k=31((k+3)−3)4[Drop the index from k=6 to
k=3]=∞∑k=31k4=∞∑k=11k4−114−124=π490−1716
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