4.* Let a > -1 be a real number. Prove that (1+x)" > 1+nx holds for any integer n >0 (this is called Bernoulli's inequality). Be very careful to see in your proof where you needed that r> -1.

Q: 5) Prove that : tanz 올n(를) i-z

A: tan-1z=i ln[(i+z)/(i-z)]/2.

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Q: Whenever n is a pOsitiv integer (n > 1). (15 points) %3D

A: Follow the procedure given below.

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Q: Prove that the sum of (") where i is from 0 to n is 2".

A: According to the binomial theorem, 1+xn=n0x0+n1x1+n2x2+⋯+nnxn……1

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A: We have given that a matrix k of order 3*3.

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Q: 2. Show that In (- - ) = ( + 2nn)i where n = | .... .... ..

A: Consider the following expression: x=ln-12-32iex=-12-32icosx+isinx=-12-32i

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A: To prove that  L(cosh(kt))=ss2-k2

Q: 6) Prove that ni 7n² for every integer nz4,whereas. n 7 n3 2 farevery integer n$6.

A: We have to proof by induction process

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A: We know that ln(1-x)=-x-x22-x33-x44-...=-∑n=1∞xnn     ...(1) on the interval (-1, 1)

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A: Evaluate the given function  as follows.

Q: See Image for the Question

A: Here, the idea is to write xy - x0y0 in a way that it becomes easy for us to solve.

Q: 3)suppose fle) isadifferentialile funehou on the intervall E-7,1J Such that l-7) =4 and f6)=1.…

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Q: If n>1 prove the sum %3D is not an integer. 1无

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A: We have to fill blanks from given statement Clearly to guarantee at least six people are literate,…

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A: Fermat number.

Q: Least residue of 45^37 mod 19 when using Fermats Little Theorem

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Q: 6. Verify Euler's Theorem for n = 16 and a = 5. (7 from 6.5)

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Q: 27. Show that if x and y are positive real numbers that have the same first n digits in their…

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Q: : a number is even  if

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A: We need to express the infinite continued fraction 2, 1, 5¯ as a quadratic irrationality. The…

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A: Recall the following. Pythagoras theorem: For any right triangle, the square of the hypotenuse is…

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Q: 3. Show that all the irrational numbers in the interval [0, 1] is dense there.

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Q: 1- 9/if it of Prove that yś(-)- TT

A: In the question it is asked to prove y'(0)=π for the given function y=2xcos-1(2x)-1-4x2.

Q: 18. Show that Σkind(k)³ = (Σk|n d(k))² for all positive integer n.

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Q: Show how to adapt the proof that v3 is irrational (from the video) to prove that v15 is irrational.…

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Q: 17. prove that Edln H(n/d)7 and σ(n ) σ(n) Σ %3D

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Q: Find th: units digitof 3" ++ 11* ", whe re e and n: arc: nan-npitve intrgers

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

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