Q5. A. Prove that, if g is multiplicative function and G is defined by G(m) = Eatm 9(d)then G is also multiplicative.B. Write the Mobius Inversion formula for o(n) and verify it for n = 42

Given: Let us prove that G is multiplicative if g is a multiplicative function and G is defined as…

Answered: Q5. A. Prove that, if g is…

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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The Intermediate Value Theorem can be used to approximate a root. The following is an example of binary search in computer science. Suppose you want to approximate /8. You know that it is between 2 and 3. If you consider the function f(x) = a? – 8, then note that f(2) 0. Therefore by the Intermediate Value Theorem, there is a value, 2 < c < 3 such that f(c) = 0. Next choose the midpoint of these two values, 2.5, which is guaranteed to be within 0.5 of the acutal root. f(2.5) will either be less than 0 or greater than 0. You can use the Intermediate Value Theorem again replacing 2.5 with the previous endpoint that has the same sign as 2.5. Continuing this process gives a sequence of approximations rn with T1 = 2.5. How many iterations must you do in order to be within 0.00390625 of the root?
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The Intermediate Value Theorem can be used to approximate a root. The following is an example of binary search in computer science. Suppose you want to approximate /8. You know that it is between 2 and 3. If you consider the function f(x) = x? – 8, then note that f(2) 0. Therefore by the Intermediate Value Theorem, there is a value, 2 < c < 3 such that f(c) = 0. Next choose the midpoint of these two values, 2.5, which is guaranteed to be within 0.5 of the acutal root. f(2.5) will either be less than 0 or greater than 0. You can use the Intermediate Value Theorem again replacing 2.5 with the previous endpoint that has the same sign as 2.5. Continuing this process gives a sequence of approximations xn with x1 = 2.5. How many iterations must you do in order to be within 0.015625 of the root?
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Q5. A. Prove that, if g is multiplicative function and G is defined by G(m) = Ealm g(d) then G is also multiplicative. B. Write the Mobius Inversion formula for o(n) and verify it for n = 42