Let a > 0. Show that the series E(1+ a")-1 is divergent if 0 < a <1 and is convergent if a >1

Q: . Find the values of p for which the series E=1 Inn is convergent. (with reason)

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Q: | Find the Maclaurin infinite series for the function : y(x) = sinh (n² x)

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Q: Let 1 f(x) = 2 – 3x Write the power series about x = 0 for f(x), and find the interval of…

A: We have to find the power series and interval of convergence for the given function about x = 0…

Q: Using the nth term test for divergence, the series 2n n +5 an n31 n-1 is divergent since lim an =

A: According to nth term test for divergent, The given series is,

Q: Suppose that a power series f(x) = L a„x" n=0 has the radius of convergence p > 0 and that L.- anp"…

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Q: Find the positive values of x for which the series > xnn is convergent. n= Is the function f(x) = >…

A: Consider the series ∑n=1∞xlnn We have to find positive value of x for which the given series is…

Q: Determine the convergence or divergence of the series. E sin( ( (2n – 1)m 2 n = 1 converges O…

A: First simplify the given expression and then we can easily say that series is convergence or…

Q: (a) Consider the series For which values of a > 0 does it converge and for which does it diverge?

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Find a power series representation for the following function and determine its radius of…

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Q: Find a power series for the function f(x) = 2/(5x + 4), centered at c = -1, and determine the…

A: Given Data The function is f(x)=25x+4. The center at c=-1.   The expression to evaluate the power…

Q: Find a power series for the function f(x) = 5/(4 − x2) , centered at c=0 , and determine the…

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Q: The series E is (i) Diverge (ii) Converge

A: The given series is: ∑n=1∞1n45 Which is of the form ∑n=1∞1np And p-series converges if p&gt;1 and…

Q: Find the interval of convergence for the series E- nx" in=D

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Q: The series E² is convergent and has a sum.

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Q: 2 Σ e -n=1 In +1)

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Q: Give an example of two series E an and Ebn, both convergent,such that anbn diverges.

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Q: a) Find a power series for f(x) = %3D 1-8х b) Give the interval of convergence for f(x)

A: Hey, since there are multiple subpart posted, we will answer first 2 questions. If you want any…

Q: Let E an be an absolutely convergent series with an # 0 for all n. Show that sin(an) n=D1 is also…

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Q: Find the taylor until 4 non zero terms for series a) f(x) = In(x) at X= 3

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Q: Find all values of x for which each power series converges. Σ •a. (x +2)" l=ח

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Q: Let x - 1 f(x) = r +2° Find a power series for f and determine the interval of convergence.

A: Finding the power series for f(x) and find the interval of convergence

Q: Use sigma notation to write the Taylor series about x = 7 for the function. 1 x + 5

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Q: Find a geometric power series centered at e = 0 for the function: 2.x h(x)=- x' -1

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Q: Let L = lim |an]/n The series an n=1 converges if

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Q: The radius of convergent for the series (a)co (b))

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Q: Use power series to evaluate the limit. lim as x goes to 0, (e^x-(1+x))/x^2

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Q: Find the Laurent series that converges for 0 < |z – zol < ∞ with zo = "/4 for the function sin z 3

A: Given function is fz=sinzz-π43

Q: Suppose that > an(x+9)" converges at a = 13. At which of the following values of x must the series…

A: Consider the given information.

Q: Let f(x) be a differentiable function such that for all x > 1, f(x) > 1 f(x) and f(x) < lim f(x) = 1…

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Q: Find the first five non-zero terms of Taylor series centered at x = 1 for the function below. %3D…

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Q: The series ∑Cn(x+9)^n converges at x=−4 and diverges at x=−17.

A: We know that if the power series ∑ n=1∞an (x-b)n is convergent at x=a, then this series is…

Q: Let 2 ak converge with the positive Ak Converge with the positive term- In this case examine. the…

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Q: Find a power series representation for the function and determine its radius of convergence. f(x)=…

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Q: 1 f(x) = (1+x)²

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Q: If E an is a convergent series with positive terms, is it true that the series ( a) Σ(1+ α) (b)…

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Q: write the function as a power series using sigma notation, and find its interval of convergence…

A: Known fact: 11+x=1-x+x2-x3+⋯  provided x&lt;1.

Q: Find the interval of convergence of the series E*. k=1

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Q: If E a, is a convergent series of positive terms, prove that An is convergent. 1+ an

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Q: 3. If the series Eo akx* converges to the function f(x) on the interval (-2, 2), provide a formula…

A: Given:  So, the nth derivative is:

Q: Define Find the values of x where the series converges and show that we get a continuous function on…

A: The series is  . All the   are even and that's why consider .

Q: Use sigma notation to write the Taylor series about x = 3 for the function. ON X 1

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Q: Using the nth term test for divergence, the series Σ 32n+1 an = n=1 n=1 is divergent since lim an n→…

A: We need to evaluate given limit

Q: Consider the following function, defined by a power series: f(x) = 2n 1 n=1 Determine the radius of…

A: Given a function, fx=∑n=1∞xn2n-1. The given function is of the form, fx=∑n=1∞anxn. The radius of…

Q: Find a power series for the function f(x) = 6x/(x2 + 4x − 5'), centered at c = 0, and determine the…

A: Solution: Consider the given that the function f(x) = 6xx2+4x-5orf(x) = 6x(x+5)(x-1) Now making…

Q: Using the nth term test for divergence, the series a, = 32n+1 n=1 7n=1 is divergent since lim an=

A: "Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Find a power series representation for the function and determine the interval of convergence: f(x)…

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Q: The series E "k 1 is ek 3D1 absolutely convergent divergent conditionally convergent no conclusion

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Q: Let fn(x) be the following sequence (not series) of functions n2 fn(x) = (x2 + 7x + 6)n² +n

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