Prove the following combinatorial identities (algebraically): (") = Eo ("*) 1) ("") = E, () (") 2)

Q: Show limn→∞(1 +1/2+1/3+···+1/n−lnn) exists

A:

Q: Find the limit: 5 cos(n) (8n-1)(n+10)) lim + n- 00 in 9n2+n+3 ||

A: on solving this we get the

Q: %3D 39. Prove that lim x*cos

A: To prove the given limit

Q: :f and the functions fn are all monotone increas- If limn0. fn ing, must f be monotone increasing?…

A:

Q: Expand cos'x as a series of ascending powers of x.

A: We have to expand cos2x as a series of ascending power of x. Recall that:           cos2x =2cos2x -1…

Q: 2+ 1 ** Evaluate lim - X+ oL dt A B none of these 1- e? 1 4

A: Given that,  Now taking the integral term,

Q: Prove that lim x² cos(1/x²) = 0. x→0

A:

Q: Present a “DELTA – EPSILON" proof, proving: prove th the de lim (3x – 7) = 5 X-4

A:

Q: 1. Find the Fourier series of the function f(x) = |x| which is defined over the range - T<x< πT.

A:

Q: Let bn : n° +n+6 for all n ɛ N. Find limn→∞ bn- 5n3+4

A: Given that bn=n3+n+65n3+4 and we are required to calculate limn→∞bn. Therefore, we are supposed to…

Q: 1) @) Prove that lim an= o if lal <1 and a ER 6) show fon any X with Ix) <1, lim nxn=0- that

A:

Q: х Prove that 1 < < sin x for all x e (0,5). - ド|N

A:

Q: 1. Prove that ƒ (2) = sin z is analytic.

A: "Hey, Since there are multiple questions posted, we will answer first question. If you want any…

Q: 1. Find the Wronskian for yi = x , y2 = x² and y3 = x³.

A:

Q: How does Abel's Identity allow us to conclude that if the Wronskian is 0 at a point t, then the…

A: To prove: By Abel's identity conclude that if Wronskian is 0 at a point t, then the Wronskian is…

Q: - prove that = -Gtx + c 1- cosx +'x

A: To prove: ∫21-cos2x+sin2xdx=-cotx+c To prove this, take its left side and integrate after…

Q: 21. Find the Fourier r fe -1<x <o - -IN X

A:

Q: 2 for al 2 0. 39. Prove that lim x* cos %3D -

A: Given,

Q: Which of the following situations is very much useful in proving identities

A: To answer the given question about identity.

Q: 83 Expand log (cer) Theosem in poue og (x-2) wing Taylor's

A:

Q: in a restaurant th e is continuous.

A: discrete random variable: 1) The probability of a discrete random variable lies between 0 and 1  2)…

Q: Prove that lim vn+1= 1. n-00

A:

Q: Prove or provide a counterexample to the following if r ∈ ( 0 , 1 ), then lim n→∞ r^n exists.

A: limn→∞an=L if for every number ε&gt;0 there is an integer N such that                   |an-L|&lt;ε…

Q: 19 , let e > 0 be arbitrary. Determine & as a function of e In formally proving that lim + 8 2

A:

Q: 4. Prove the formula below using mathematical induction log 2k u + zu log 2 k=1

A: Given: Formula: ∑k=1n log2k=n2+n2log2

Q: Extend the result proved in Example 2.5.3 to the case |b| &lt; 1;that is, show lim(bn) = 0 if and…

A: We have to show lim(bn) = 0

Q: 2. The limit of the following function equals: x2/3 + x-1 lim x--00 x2/3 + cos2 x

A:

Q: Consider the elliptic curve group based on the equation where a = 10, b = 8, and p = 11. In this…

A: As per the question we are given the following elliptic curve : y2 = x3 + 10x + 8  mod 11 And we…

Q: Suppose &lt;let A= 1 and you know all the cofactors in C. How can you find A?

A: Given: Determinant of a matrix A  =  1 and all the co factors are known.   To find: The matrix A.

Q: Question 11 Evaluate the limit lim (ln(3x² + 2) - In(x² + 1)) O ln 3 O In 3 01 00 O ∞

A: To evaluate the following limit.limx→∞ln3x2+2-lnx2+1

Q: For a periodic function of period a, its Fourier series will only contain even terms; that is, az =…

A: Given: ak=bk=0 whenever k=2j+1

Q: 1. Show that for all f e L'(R) lim

A: To show: limδ→1∫f(x)-fxδdx=0 Concept used: ∫fxdx=Fx+c Chain rule: ∫fxadx=1aFx+c

Q: e -e lim is z0 In(1+ 2x)

A: Given limit:

Q: Example 12 Given the function f (x) = In cos x. Obtain the Maclaurin's expansion for f (x) till the…

A: We’ll answer the first part of this question since due to complexity. Please submit the question…

Q: 3. Expand cos'x as a series of ascending powers of x. OS

A: To expand cos^2x as a series of ascending powers of x.

Q: 5. Find the term representing the third har- monic for the periodic function of period 27 given by:…

A:

Q: please help thank you

A: We need to prove the given statements, which are,

Q: Let an for all n e N. Working directly from the Vn+1 definition show that limn- An = 1.

A: To evaluate limn→∞ nn+1: Here, an=nn+1 an can also be written as: an=1n+1n Simplifying this we get,…

Q: Evaluate the limit: limo* [cos(3x)]= O e-3 03 e 01 ◄ Previous

A: To evaluate the limit.

Q: How does Abel's Identity allow us to conclude that if the Wronskian is 0 at a point t, then the…

A:

Q: Find the complex Fourier Series for periodic extension of: x(t) = 2sin(t) cos(t) For 0 &lt; t &lt;…

A:

Q: 5. Find the term representing the third har- monic for the periodic function of period 27 given by:…

A:

Q: 17. Prove thatsin x = =x 1-x2 dx

A:

Q: In(x), 5 s x s 16 f(x) A = lim n- o 2 i = 1

A:

Q: 2 39. Prove that lim x"cos = O. X-0 X

A: The given limit to be proved is

Q: 3. Prove the following using the epsilon-delta definition of a lir a. lim (2x – 1) = 3 x→2

A: We will prove the limit using definition.

Q: 16) [5] Compute the value of the following limits using L'Hôpital's Rule: Ax -1-4x a) lim x40…

A:

Q: 6) Use the first three terms of the Taylor expansion for f(x) = cos(x) to estimate cos(1/8). %3D

A: We have to find the value of cos(1/8)using only first three terms of Taylor's series.

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.