m îs a positive intege; p=Lim +1, a prime number; be a positive integer nd dim, In this Let d ose Prove it,

Q: Q: Apply Taylor's expansion and find at The seyies for tenx 4 (edo:3)

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Q: Let a1 = 1 and an+1= √ (an+2). Prove that a nis convergent and find limn→∞ an

A: we have to find limit

Q: 6) The Ying of integers(2) is ideal of veal number (R) an DTrue D Fulse

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Q: IF TE A (V) and if 9 (1) € Fcx) is such that 9 (T) = 0, a characteristic root it true that every…

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Q: Q: Use the Squeeze theorem to estabich cennergence of the gi , ren Sequence Conr Inx X(X+2)

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Q: O use (aflace transform to frnd charge ace) at fime t in civwit shoun en following fgure, if R=l009…

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Q: E an diverges and lim an/bn = 0, then b, diverges. n=1 If n00 n=1 True False

A: Limit comparision test

Q: If D is an integral domain and if na = 0 for some a is non zero in D and some non zero integer n,…

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Q: 2) Let m and n be positive integers. Use the Convolution Theorem to evaluate the integral…

A: Let m and n be positive integers. The objective is to evaluate the integral ∫0t(t-τ)mτndτ by using…

Q: Prove that At 1l ellse

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Q: omot rix over the field F& roo- Every equitha lemt to mx n veduced metoix.

A: Given a m×n matrix.

Q: 1 For any n N set In = * cos" (t)dt, known as Wallis integrals. 1. Show that (n + 2)In+2 = (n+1) In.…

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Q: Define for x>0 the function dt In x = 1. Show that xH In x is differentiable on (0,00). 2. Show that…

A: We use the definition of differentiability to check the function is differentiable.

Q: Assume there is a sequence of real numbers (tn)=1 such that lim t, = 0 and f(tn) = g(tn) for all…

A: The two power series functions given are as follows: fx=∑k=0∞akxk    and    gx=∑k=1∞bkxk We are…

Q: By Find the Taylor senies for fixl= sm x Compating deriratives, at x=0.

A: We know that, Taylor series for fx=sinx at x=a can be found as Now, The Taylor series around a=0 is…

Q: Suppose that E is an extension of F of prime degree. Show that, forevery a in E, F(a) = F or F(a) =…

A: Let E be an extension of F of degree p (prime) => [E : F] = p Here we will use tower property to…

Q: Prove that for every v = (x, y) E R? there exists a sequence v, = (xn, Yn) € R2, where rn, Yn are…

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Q: Let (r.) be a sequence of positive numbers with Intl = L. lim n0 In Let 0 <e< L. Show that there…

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Q: 2. Recall that the characteristic function of a set E is defined as I is in E 1, Xe(z) = { 0, z is…

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Q: Q2: Let R a, b, c E Z}and let p: R → Z be defined such that = a. 4. Show that ker () is a prime…

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Q: SHOW by Induction that : Alx)" /1-(1-2x)" A For all n e IN" from A (aS is a matix) obseivation Where…

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Q: f(x) is a functicn given by Icosxl Assume that floe) find the pourier Series for of fe).

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Q: Suppose m is a positive f& La(μ), 11 £1100 2o and 2n = Ln Prove that = √ 1+1" 1₂" (n=1₁2,...)…

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Q: Find the expansion of Fouris seies for the fomning functlon. 人メズ -- x 60-

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Q: Prove for uniform covergence the series 1 „VxeR. (By using Weierstass's M-test). 5 7 2 n=1 n" +n'x?

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Q: (b) There exists LE R such that L4 Size(O(n)).

A: Given below clear explanation

Q: A pply Taylor's s expansion and find the se yies for tann at = 4 %3D

A: We need to find Taylor's expansion for tan2x at π4.

Q: Let us expand f(x) = x, 0<r < 1 in the series f (a)-ΣΑ, J, (μεα) , k=1 where denotes the kth zero of…

A: Given : fx=x3 , 0&lt;x&lt;1 We have to expand fx in Fourier-Bessel series           fx=∑k=1∞AkJ3μkx…

Q: Evaluate An+1 lim for n! An nn and determine the serie Σ n=1 converge or diverge? 1/e and diverge e…

A: We have to check

Q: If (an) tends to infinity and (b,n) is bounded, with bn > 0 for all n, then (an/b,) tends to…

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Q: 28. Let (п — 1)2 12 S, 22 ... n? To calculate lim,2 S,, show that Sn + ... + п п and interpret S, as…

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Q: Using the nth term test for divergence, the series 00 08 An > V42n+1 %3D n=1 is divergent since lim…

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Q: Suppose a a. (Hint: %3D

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Q: Let dg denote the number of derangements of order k. (a) Prove the identity C) d = n! with a…

A: Given dk is number of derangement of order k. We know that for a set 1,2,…,k, the derangement dk are…

Q: Determine the following by using an appropriate Riemann sum. lim ... + n] 1? + 2 + 32 + n → 0 n-

A: Limit.

Q: Let {xn} and {yn} be sequences such that lim yn = 0. Suppose that for all k ∈ ℕ an for all m ≥ k we…

A: Let {xn} and {yn} be sequences such that lim yn = 0. Suppose that for all k ∈ N and for all m ≥ k we…

Q: Compute the following limits. 1. lim nsin(2ren!). Hint: use Taylor expansion with the Lagrange…

A: The limit is equal to 2π.

Q: 4. From the field of (Z5, , (mod 5)), obtain two MOLS of order 5.

A: When p is a prime number, the field ℤp is the set ℤp=0,1,2,3,....,p-1. Over the elements of ℤp, a…

Q: Choose {X}1 from [0, 1]. Let pn be the probability that X + X4+1 <1 for all i e {1,2,...,n – 1}.…

A: Given that Xi for i = {1,2,3,...,n} are distributed uniformly from [0,1]. Let us consider two random…

Q: Let A ∈ Mn×n(F). Under what conditions is det(−A) = det(A)?

A: In determinant the scalar quantity is taken common from a single row or a column.

Q: Let こ ニ be the Square with Sides of length 2. Let In be the Square pimed at Co,0) ratated through an…

A: Given,I=(x,y): x≤1,y≤1 : SquareIn is rotated at an angle 2πnθTo find:  limn→∞sup (In) and limn→∞inf…

Q: Ler Flオリ= J a te defiattion of fayler sevies to find the taylor sener of'f"when x- in sigma notation…

A: Given f(x)=1x at x=14

Q: lim n→∞ n² n² 9n² +10n +3 = 9.

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Q: Determine which formula(s) can be used to find the pseudoinverse of A = A+ = AT (AA")-1 O At = (A"…

A: Known fact: IF the columns of a matrix A are linearly independent then pseudoinverse of A is given…

Q: Applying the ratio test to the series ∞ 1 k you would compute ak+1 lim = lim k→∞ k→∞ ak k=1 11

A: Given query is to apply ratio test.

Q: a) Prove that the series f(2) = Ee -" sin(nz) n=1 defines a holomorphic function in the strip -1 <…

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Q: Let ain be integers with 2<n ond ecd (ain) - d. Show that if ok #I lmodn) for all k E { pln)-} then…

A: We know that Uℤn is the set of all units of ℤn. First let k=1, then ak=a1=a. Given that gcda,n=1.…

Unitary Method

The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products. 

Speed, Time, and Distance

Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.

Profit and Loss

The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.

Units and Measurements

Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.