Show that os fj ex* +en+ 22n+1 + (2n+1)n! 25 e*(22n+3) dx – |2 + ... 5(2!) (n+1)!

Q: a) Obtain power series solution in powers of x for (x2 – 1) y" + 3x y' + xy = 0, y(0) = 4, y'(0) = 6

A:

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: 6. Consider the integral f(n + 1) = x"e¯* dx where n20. CGive a convincing argument that S* x"e¯*…

A:

Q: 1 (a) Use the Integral Test to show that > converges. 2. n' +1 d 1 n = 0 -1 tan dx Hint: 1+x?

A: As per our guidelines, we are supposed to solve only one question.  Kindly post other questions  as…

Q: In(x²+x)-In x+eln 4 dx In e(x+1)

A: Given integral, ∫lnx2+x-lnx+eln4ln ex+1dx

Q: a random varrable Y has a cumdative df fy if 04 >4 probability hat Y falls between 2land 3 find thd

A:

Q: Q4. Test for convergences of the following: 7 図t X 2n=2 10-1 Vn+1-Vn 131 Vn2+n

A:

Q: 5. Let fn(x) ". Show that z"0 uniformly on (0, ).

A: We have to use squeeze theorem here.

Q: The series 2" (n!)2 n=1 is divergent. Select one: True False

A: The given series is a 2n/(n!) 2. Repost the next question.

Q: If the têst is inconclusive, then say SO. 3* Σ k! k=1 If p = +o, enter "inty". i

A: Given query is to find series converges or diverges.

Q: 3, Re call Eudlid's proof of In finitude * primes! Shouw that T Cx): = 1{ ps X: p-pinae }12 logz…

A:

Q: ii. Investigate convergence or divergence of 2n + 100 3n +1). n=2

A: Given series is  ∑n=2∞-1n2n+1003n+1n

Q: 9 Open on a Jamboard Set background Clear frame ha: y= Va: X = 4 5 Domain hole: none X-int: axti= 0…

A:

Q: (c) Show that fn (x) : x2+n? converges uniformly on [0, 0).

A:

Q: Discuss the Convergence or Divergence for D { [4+ (-0"]" (Z+2)" こう 2n+1 - DZ² 2 لع Wi ngo (2n+¹)!

A:

Q: Find the interval of convergence of E 27" (x – 7)3n+2 n=0

A:

Q: Suppose that 0< fn < In on X for all n e N. If gn 30 on X, then fn 30 on X.

A: Solution:-

Q: Show that the series Ln=0(2-1 1 6) converges uniformly on (z: Iz| 2 2).

A: Consider the given equation,

Q: Find the probability P (0<x<) for f(x) = x². cos x³ 0.235 O 0.304 O 0.908 O 0.413 O 0.678 O 0.112 O…

A: We need to find tge probability P(0 &lt;x &lt;π/3) for f(x) = x2 cos x3

Q: Q3: Show that 3(2)n+2 3(2)n+2 is convergence and En=0 5n Zn=0 5n 20.

A: Please refer to the image below.

Q: 99. Show that 00 cos x dx is absolutely convergent.

A:

Q: 10. Find the integral of dx x²-4 O In() + C x-2 O In() + C 2x x+2 O fIn( *2) + C X+2 O In() + C x+2…

A: *using concept:- 1. Partial fraction. 2. int(dx/x) = ln(x). 3. ln(m) - ln(n) = ln(m/n).

Q: cos(nT) n=D1 converges conditionally in Select one: O True O False

A:

Q: 1. (a) Prove the following. | (In x)"dx = x(In x)" – n (In x)-' dx

A: To prove that 1.a. ∫lnxndx=xlnxn-n∫lnxn-1dxb. Using 1.a compute the following      ∫lnx3dx

Q: Proceed as in Example 3 in Section 6.1 to rewrite the given expression using a single power series…

A: The answer is given as follows

Q: In 7 Perform a test to detemine if 7x-2 e e-71×dx converges. In 7 |7x-2 e -71x dx = (Type an exact…

A: Given integration is : ∫0ln77x-2e-7xdx To find the value of integration.

Q: Q4. Test for convergences of the following: 7 因 + 2n=2 10n-1 Vn+1-vn X En=1 Vn²+n

A: a∑n=2θ710n−1Using Ratio TestL=limn→∞an+1aniIf L&lt;1,then the series converges absolutelyiiIf…

Q: 5.6) Apply the integral test to the series 1+e2n n=2

A:

Q: 2. Find the series solution of the equation xy" + 2y' + xy = 0 in a form: y(x) = y(x)=(ao )=²(ao c…

A: The given problem is to find the series solution for the given differential equation. The given…

Q: Find the recurrence relation for the power series solution about x= 0of the differential equation y"…

A:

Q: 8n 4n2 n = 1 + 1 f(x) = converges O diverges

A: Given query is to find integral converges or diverges.

Q: Find the probability P (0<x<¹) for f(x) = x². cos x ³ 0.235 O 0.304 O 0.908 O 0.413 O 0.678 O 0.112…

A: Solution

Q: En=1" Vn sin а. 1 b. 2 c. Converge

A:

Q: 1. Let Sn Discuss the convergence of {Sn}=1 and En=1 Sn n+1

A: The given sequence is sn=nn+1. Compute the limit as follows.…

Q: What is the power series representation of the derivative/integral (whichever you chose in problem…

A: (3).  The given function is: f(x)=lnx2-4

Q: d Prove thate* = e* from the first a %3D dx principles only definition, without using natural logs…

A:

Q: determine the radius and convergence of E n=1 5xn/3n2

A:

Q: Using the power series method about x = y" + xy' + y = 0. 0, the recurrence relation of the…

A: Compute the first derivative of y as follows.

Q: 6. Find at least the first 4 nonzero terms in a power series solution about the regular singular…

A:

Q: Determine the region of convergence for n° + 2n° +1 (z+1)" 4n +9 100 Σ 2n2 +9 3" (n² + 10n+ 5) (z-…

A: We have two infinite series, from which the first one is sum of finitely many terms, hence the…

Q: 3. Prove that So 0 1 x + ex -dx = n=1 (−1)n−1(n − 1)! nn

A:

Q: If C(x)=∞∑n=0 2n/4n^2+1 x^2n and S(x)= ∞∑n=0 2n+1/(2n+1)^2+1 x^2n+1, find the power series of…

A:

Q: If |Xn+2 − Xn+1| < |¤n+1 − xn| for all n E N then (xn) converges. -

A:

Q: 5) Find the power series solution of the ODE given by y" –xy =0 in the form y = Cx where y = y(x).

A:

Q: 5) Find all values of x for which En=1 nxn (n+1)(2x+1)n converges absolutely.

A:

Q: 18. Show that Ein d(k)³ = (Ekin d(k))² for all positive intege m

A:

Q: In x + In(x – 1) = 10 In(x + 25) – In(x – 1) = 3 e2x – 2ex = 15 e2x – ex – 12 = Q %3D %3D -

A: i) ln x+ln(x-1)=10ii) ln(x+25)-ln(x-1)=3iii) e2x-2ex=15 Note:      Multiple part question are given,…

Q: 4. Evaluate the following integral fel e(-²²-25) dt

A: Note:-  ∫ e-x2 dx=π2 erfx +C   Given integral is ∫0∞ e-t2-254t2 dt

Q: 3+ Evaluate the integral: e³+ cosh2x33+ cosh2xsinh2x dx (3e)3+ sinh2x A +C 2+In3 (3e)3 + cosh2x +C…

A:

Q: 1 i. Use the result in (i) to test for convergence 2n =1 2 Vn? +5

A: See the attachment