Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive. P = n an+1 n→∞0 an = lim 1 (2n)! is: = O A. convergent O B. divergent OC. The Ratio Test is inconclusive (Enter 'inf' for .) 1 (2n)!

Q: 3. Define fn on (0, 1) by fn(x) = x". Show that fn converges to the constant function 0, but that…

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Q: Prove from first principles (i.e. give a direct e-N proof) that the following se- quence converges:…

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Q: n cos(n) V1+4n2 (an) at n→∞o or prove that the limit does not exist: (b) In(1+2n²) - 2 ln(1 + n)

A: 7) a) an=ncos(pi n) /sqrt(1+4n2)b) an)-2ln(1+n)

Q: 6. Find the interval of convergence of -x* . Where p >0 is a fixed constant.

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Q: 24.1 Let fn(x) = nx 1+2 cos² nr. Prove carefully that (fn) converges √√n uniformly to 0 on R.

A: We have to show that  converges uniformly to  on

Q: 6.3 Represent the function f(x) =,, using a power series and find the interval of convergence. 2 – x

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Q: 3. Prove that the series 1 x2n n! 1 + x²n n=0 converges uniformly on R to a continuously…

A: We'll follow these steps:Show that the series converges uniformly on R.Prove that the limit function…

Q: Show that Show that Show that fexx dx converges for any p = R. In xdx and f′ (x(ln x)(1 + x²)−¹)dx…

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Q: 2) Suppose that f(x) = 5x² over the interval [1, 2]. Let Ax = 1/n. Let x₁=1+iAx. Find the limit of…

A: Given: fx=5-x2.Formula to be used: ∑i=1ni=nn+12 ; ∑i=1ni2=nn+12n+16

Q: Prove by contradiction the scquence {(-1)"+} is divergent.

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Q: Let fn(x) = sin" x and let 8 > 0. a) Prove: fn converges uniformly on [0, 5 – 8], but not uniformly…

A: The given function is fnx=sinnx. To find: (a) fnx=sinnx uniformly or not on given domains.…

Q: Suppose (xn) does not converge to 0. Then there exists e>0 such that all except for finitely many…

A: Consider the sequence xn=1,0,1,0,1,0,....... Here the odd terms are 1 and the even terms are 0.…

Q: 1 √x+x4 So dx converges.

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Q: 6. Consider the sequence of functions n²a* – sin(nx) fn(x) = x E (0, 00) n2x (i) Show that fn…

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Q: Find the interval of convergence for the series f(x) z£ 3(x – 2)".

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Q: Suppose f(x) > 0 and g(x) > 0 and that | f(x)dx diverges. If f(x) > g(x), what can we conclude about…

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