3. Let {an} be an increasing sequence of positive real numbers such that the series ak is divergent. If {sn} k=1 be the sequence of partial sums of =1ak and tn = とk=2 Sk- n = 2,3, 4, ..., then find lim,n→ tn, if it exists.

Q: Σ | 6.8 Differentiate the series 2 (n + 1)x" term-by-term to find a power series representation for…

A: We need to obtain the series for 2/(1 - x)3 from the series for 1/(1 - x)2

Q: (9) Suppose that ) An is a series so that its partial sums Sk = a1 + a2 + ...+ ak satisfies n=1 1…

A:

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A: Taylor series expansion for the function f(x) at  is x=a.

Q: Prove that if n > 1, the nth partial sum of the harmonic series is not an integer. Hint: Let 2* be…

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Q: п(п + 1) Compute the general term an of the series with the partial sum Sn -, п> 0. an

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Q: 3. Use the Comparison Test to determine if the series 5n Cos 2n - n³ + n² + 1 n=1 converges or…

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Q: 4. Let {ak}1 denote a sequence and let {sn}, denote the corresponding sequence of partial sums for…

A: The solution are next step

Q: If n'h partial sum of the series i=| ak k31 is s, = ( 3n-1)2n %3D 3n then Li=1 ak =

A: The nth partial sum is Sn=3n-13n2n The  we have to find ∑k=1∞ak

Q: 2. Prove that the following infinite series diverges. Σ(-1)*. k=1 (Hint: write down the first few…

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Q: 1 Find the Taylor series for f(x) = 1+2x at x = 2 and its interval of convergence. Select the…

A: First calculate interval of convergence:  fx=11+2x       =11+2x-2+4       =15+2x-2…

Q: 2. Determine the convergence or divergence of the series. n2+1 2n2+1,

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Q: 2. For the infinite series E(-1)* , write out the first four terms of the sequence of k-1 partial…

A: I am going to solve the given problem by using some simple calculus to get the required result.

Q: 1 Now compute s5, the partial sum consisting of the first five terms of k=1 85 (5)5

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Q: Expand f(2)=z(z - 4) 1 < |z-51 < 5 in a Laurent series valid for the indicated annular domain. (Give…

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Q: 1 - (a) Find the series of valid in the domain 0 < |2| < 3. z²(z + 3)

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Q: If the nth partial sum of the series 2n²+3 E an is Sn = in=1 3n²+1 then Ln=1 an

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Q: expand function f(z) into the laurent series at point z0 for given intermediate circle: f(z) =…

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Q: Suppose that an is a series so that its partial sums Sk = a1 +az + ...+ ak satisfies %3D n=1 1 1.…

A: In kth partial sum, Sk of an infinite series converges to s, then series also converges to s. Also…

Q: Find the Taylor series for f(x) centered at 5 if 8(n+4) f\n)(5) =

A: Introduction: The Taylor series of the function f(x) centered at x=a is given by,…

Q: 1 (b) Expand f(2) = in a Laurent series valid for 1 < |z| < 2. 22 – 3z + 2

A: The given function is fz=1z2-3z+2

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Q: 1 Expand function f(z) = in a Laurent series at z, =0 and determine its az + b convergent area,…

A: We will use the fact that  1/(1-z)= 1+z+z2+z3+.......

Q: Expand f(z)= in a Laurent series valid for 2 <z <3 (z+2)(z+3)

A: The solution is given as

Q: Find the interval of convergence of the series. (x - 2)2n Σ 36 n=0 1<x < 3 -4 < x < 8 X < 8 -8 < x <…

A: Given: ∑n=0∞x-22n36n

Q: 1 Find the first five non-zero terms of the Taylor series for f(x) = - X Written compactly, this…

A: Taylor Series for f(x) centered at x=a is given by f(x) = f(a) + f'(a)1!(x-a) + f''(a)2!(x-a)2 +…

Q: 3. Use the Comparison Test to determine if the series 5n – cos 2n - n3 + n² + 1 n=1 converges or…

A: Use the comparison test to determine the series.

Q: Given series is > =£ 2n +5, Consi der a, 3n-1 2n +5) 2 3-. 3n -1 2n +5 3n -1 3-0 3 Now L= lim a, =…

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Q: 2. Consider the alternating series 1 –+-+..., i.e. Eak where az = (-1)k-1 for all k e N. 4 Let Sn…

A: S = 1-12+ 13- 14+ ..... = ∑kak   where ak = -1k-11k  ; k∈ℕ.Sn =∑k=1nak     is  Partial sum.…

Q: The series > n=1 1 (a) converges because lim = 0. n0 Vn (b) diverges because lim = 0. (c) converges…

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Q: 8z +1 in a Laurent series valid for 0 < |z| < 1. (Ъ) Еxpand f(z) — z(1 – 2)

A: The given function is  f(z) = [8z +1]/z(1-z) 0&lt;|z|&lt;1

Q: Suppose the infinite series > am is divergent. Then for any real number c, the series ca, is also…

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Q: b) Use the result in part a) to find a closed form of the nth partial sum of the series

A: (30)k = (6k+1 - 5k+1)5kA + (6k - 5k)5kB

Q: Prove that if n > 1, the nth partial sum of the harmonic series is not an integer. Hint: Let 2* be…

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Q: 4. Consider a power series of the form E anx". Suppose that lim sup Jan|/n = 0. Prove that the…

A: To apply root test, we transform given series to new series where x is taken arbitrary.

Q: 8. Suppose that for sequence of real numbers {an} and {b,} that O< bn+1<an < bn, Vn EN Prove that…

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Q: Suppose that the nth term of the sequence of partial sums for a given series is Sn = 1+2n+3n² 4n²…

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Q: Suppose the series a; has a sequence of partial sums given by sn = 1+ n²–1 2n2+3' Determine i=1 if…

A: Given that ∑i=1∞ai has a sequence of partial sum as sn=1+n2-12n2+3. A series converges means its…

Q: 8. For two sequences {an} and {b,}, let cn = E arbn-k. (Then cn is the sum of the terms on the nth…

A: Please check the answer in the next page

Q: n + 2 2- Compute the general term an of the series with the partial sum Sn -,n > 0. 2" an If the…

A: Here we have to compute the general term an of the series with the partial sum        Sn = 2 -…

Q: Let c and d be positive integers. Suppose that A (In n) and B n° are the n-th partial sums of the…

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Q: (a) Determine a suitable p-series n=0 to compare with nVn+n* (b) Do specify if you are using a…

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Q: 1 Use 1- u Σ u" Ju| <1 to find a power series representation for n=0 1 f(x) = and find the radius of…

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Q: 6. Find the radius of convergence of the power series 8. Σ x2n · 4n • U n=1

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Q: expand function f(z) into the laurent series at point z0 for given intermediate circle: f(z) =…

A: Follow step 2.

Q: 14 The radius of convergence of the series > n +1 (x- 3)" is 2n +1 2"

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Q: a) Find A and B such that (12)* (4k+1_3k+1)(4* _3k) 3* A 4k _3k 3kB 4k+1_3k+1 b) Use the result in…

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Q: 3. Expand the function f(2) in a Laurent series valid for the annular region z(z – 2) given by (a) 0…

A: To find - Expand the function fz = 1zz - 2 in a Laurent series valid for the annular region given by…

Q: Determine the radius of convergence of the power series 3n (2- 5)". n2

A: Concept:

Q: 2. Find a formula for the nth partial sum of the series Σ (n+1)²) and use it to find the series' sum…

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Q: Let f(x) = 1/(x-5), and let ∞ an(x − 7)n Sigma n=0    be the Taylor series of f about 7. Then: a1…

A: Please refer the attached image for complete solution. THANK YOU.