Let denote the n-th prime number. Show that Pn Pn n=1 diverges.

Q: The sum of any two integers of the from 4k + 1is always: Od Zero Even prime

A: Sum of two integers of the form 4k+1

Q: Find the Taylor series for f(x) centered at 1 if f(n)(1) = (-1)"(n+1)! In(7)7"

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Q: 4. Recall the Fibonacci sequence: fi = 1, f2 = 1, and fn = fn-2+ fn-1. Use Mathematical Induction to…

A: The solution is given as

Q: Determine as, az, and as What is the value of b. Express q(t) as a Fourier series expansion

A: The given function is q(t)=t               for 0≤t≤π2π-t       for π≤t≤2π From the graph, it can be…

Q: Evaluate the integral |tsin(t") dt as an infinite series:

A: Given integral is : ∫t sint3 dt Evaluate the integral as an infinite series

Q: n! diverges because an > 0. for all n > 1. True False

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Q: Suppose an is conditionally convergent. Define what this means, and then use a proof by n=1…

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Q: Prove that the series E1 nis diverges.

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Q: converges, because 0 < 1 and an %3D debook 2n + 1 2n +1 2n +1 Room

A: We have to choose the true or false

Q: Determine the intervals for the following senies converaence n-1 () (5x)"

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Q: 1 Answer: diverges n In n n=2

A: If the integral ∫1∞fxdx=∞ then ∑i=1∞an diverges.

Q: Supposean converges conditionally (but not absolutely). Let A+ = {n €N : an ≥ 0} n=0 and A_ = {n € N…

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Q: Show by example that Σ(an/bn) may diverge even though Σan and Σbn converge and no bn equals 0.

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Q: R² = gen gen

A: see below the reason

Q: Let o (n) be the sum of the divisors of n and let (n) be the Euler phi functions. Let n = = pq be…

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Q: Prove by induction that (2n+1)3 is an odd number for all positiveintegers n

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Q: Assume C and D are chosen at random from {1, 2 . . . , n}. Let pn be the probability that C + D is a…

A: Given: C and D are chosen at random from {1, 2 . . . , n}. Then the number of pairs (c, d) with sum…

Q: 2. Show that if a > 0, then the sequence (n²a?e¬n*) coverges uniformly on the interval [a, ∞), but…

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Q: Q/ Drive the Taylor series for: (X — 1). е* пеar X = 1

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Q: Suppose that xo 22 and xn = 2 + √xn-1-2 for ne N. Use the Monotone Convergence Theorem to show that…

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Q: 69. Use the principle of mathematical induction to prove that dk : (In x) = (–1)k-1 «– D! for k > 1.…

A: Use the principal  induction to prove thatdkdxk(ln x)=(-1)k-1(k-1)!xk , for k≥1.

Q: Show that the harmonic series ∞∑n=11n diverges.

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Q: O Determine whether En=1 st. converges or diverges. Justify your answer by citing a 3n+2

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Q: Q/ Drive the Taylor series for: (X — 1). еx пear X3D 1 |

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Q: (a) Prove that the series Vn + 2x - Vn +x Σ n=

A: answer is in next step

Q: Show that Ekin d(k)3 = (Ekn d(k))2 for all positive integer n. %3D

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Q: test the follown divergence or 1. nin2n n=2

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Q: converges to 0 Select one: O True O False

A: It is a series with increasing powers of π/3

Q: A shop gives gift cards in nominal 25 and 40k. determine the possible total amount that can be…

A: Proof at first, the amounts seem semi-random:

Q: Use any method to deter (a) 13 The series 8.

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Q: Let the characteristic of F be n =/0 . Then n is prime.

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Q: Let C(n) be the constant term in the expansion of (x+5)". Prove by induction that C(n) = 5", for all…

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Q: 7) Show that if a, > 0, and Lim na, + 0, then the series + a4 + diverges., a, + + az 2.

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Q: Use the Root Test to determine the convergence or divergence of the ser 10) E: (In n)" n=2

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Q: Write the Taylor series for f(x) = x³ at x = -4 as > en (x + 4)". n=0

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Q: | +スt3+ *n = nintl) for all 1 てU

A: The given formula is                                    1+2+3+..........+n=n(n+1)2       for all n≥1…

Q: Is the stries cm vergent or divergent? (Ghow how you kenowi) 2 3n n'+ In + 13

A: a The given series is ∑n=1∞23n.We know that the series ∑n=1∞1n is divergent.And…

Q: Suppose an is conditionally convergent. Define what this means, and then use a proof by n=1…

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Q: Determine if the given is convergent or divergent. I In"(n*) 23n2 n=1

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Q: hlhat will be the Taylor Series oL In ! about x=0 up to and' Including the terms of order three?

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Q: Find the Taylor Series for f (x) = x° – 10x? + 6 about x = 3.

A: Use the formula of Taylor's series expansion

Q: IV. Determine if the given is convergent or divergent. s In"(n*) 23n2 #4. n=1

A: Given series is ∑n=1∞lnn(n4)23n2

Q: The series 5 n4+ 8 In=17 n++1 is divergent. Select one: O True O False

A: Given query is to find statment is true or false.

Q: Let a, be the n th digit after the decimal point in 2n + 6e. Evaluate an(.1)". n=1

A: in given problem we have to evaluate :

Q: Find the Taylor series generated by fat x = a if f(x) =i ,a = 2

A: The Taylor series formula is: f(x)=f(a)+f1(a)1!(x-a)+f2(a)2!(x-a)2+f3(a)3!(x-a)3+...…

Q: Find the Maclaurin series expansion for f(x) = xIn(1– x) Select one: a f(x)=1- X to

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Q: Represent the function 6 In(10 – x) as a power series (Maclaurin series) f(x) = Σ cnx"

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Q: 4. Prove that the L{2t²} = 4 S3

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Q: . (a) Show that if a, is a series of nonnegative terms that diverges to o, then any

A: Answer :-

Q: 2" Σ diverges n=1n99 Select one: True False

A: We have to check