2. Right-angled triangles are constructed as in the figure below. Each triangle has height 1 and its base is the hypotenuse of the preceding triangle. Show that this sequence of triangles makes indefinitely many turns around P by showing that On is a divergent series. 1 1 1 03 02 1 P 1

Q: Question 3 In(5 k) Determine whether the series >, converges or diverges. 6k a) Odiverges b) cannot…

A:

Q: 81 now that the following series is a geometric series ... 10 46 100 一

A: The series is 85+65+910+2740+81160+...The marked one are the mistakes in the sequence, have…

Q: 15. (a) Write down the sum of the geometric series z + z? + z³ + · . .+ z". (b) Hence, by putting z…

A: (a) The objective is to find the geometric series for z+z2+z3+...+zn. The formula for geometric…

Q: 1. Consider the following two infinite series: 31 2k and 24 k (a) One of these series is NOT…

A:

Q: E. Show that for 0 < p < 1, we have that Ep(1 – p)ª = 1. Hint: Remember the geometric series x=0…

A: GivenX~gem(p)P(X=x)=p(1-p)x    ; x=0,1,2,3,....we want to prove that ∑x=0∞p(1-p)x=1

Q: Again, consider the geometric series 3−1+1/3−1/9+1/27−…

A: As nothing has been mentioned whether sum of n terms is required or sum of infinite terms is…

Q: 1 The first four nonzero terms of the series of 1+I

A: A function can be represented in terms of an infinite power series by the expansion of the Taylor…

Q: 8 Let a, be the rth term of the series and the sum is defined by, n Σ n(n + 1)(n+ 2)(n + 3) ar 8…

A: The solution is given by using telescopic series as follows

Q: By writing the summation of the three power series E nCx" +EC„x"-2+ £G,"-, we obtaln: O None of them…

A: Here we will 1st convert every series in such manner that every one will have same power of x in…

Q: 4. The pattern of a sequence of functions is given by ao (x) = e-=? a, (2) = [c~*'dz az (x) = dx ´dx…

A: Given, the pattern of the sequence of functions is given as, a0x=e-x2a1x=∫e-x2dxa2x=∫∫e-x2dxdx

Q: Consider the series k0.3 k=1

A: Given series, ∑k=1∞1k0.3use the integral test we have to determine whether the series converges or…

Q: 5 Consider the geometric series 5 + +. 256 .. 4 16 64 This series can be written in the form ar"…

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Q: PROBLEM 4. Determine all r so that the series n! n20 converges.

A: Given series is  ∑n=1∞xnn!

Q: 21. For what values of p will the infinite series converge =l VI+n" OA p>3 OB p23

A: we have to use p test for convergent

Q: 42. Let e = 1+ 1/1! + 1/2! + 1/3! + Show that if k is a positive integer, then kle is not an…

A: Given , e=1+11!+12!+13!+....we need to prove if k is a positive integerthen k!e is not an integer

Q: 7. The harmonic series has the property that lim an = 0 but En=1 an diverges. Give an example of…

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Q: If ut and and u Uk are convergent series, then (uk + vx) and -) are convergent series and the sums…

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Q: 13. Find a series E ak with all nonzero terms that con- k=1 verges to 1.

A: we have to find a series ∑k=1∞ak with nonzero terms converges to 1.

Q: 27 81 Q4/ show that the following series is a geometric series ++ 46 100

A: Solution is provided below

Q: Find a formula for an for the nth term of the sequence  1/1, -1/8, 1/27, ... Assume the series…

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Q: Problem 2. (A) Prove that if the nonnegative series > an and > bn converge, then so nEN nEN does the…

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Q: Given that the sum of the first n terms of a series is given by, Sn = 2n² + 3n %3!

A: In this question, we find the nth term of the series and further we find the first term and common…

Q: (2) Prove by induction the standard formula for the sum of the first n + 1 terms of a geometric…

A: We have to prove by induction method the standard formula for the sumof the first n+1 terms of…

Q: 00 If the nth partial sum of a series ) a, is s, = 1 - n6-", Σ find an and a in n = 1 n = 1 a1 a…

A: first I have find the sequence, an , using given information and then calculate all other solutions.

Q: For an as follows, write the sum as a geometric series of the form E ar". State whether the series…

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Q: 2 Suppose that E a, is an infinite series with partial sum SN = 6 – N2 ° n=1 10 16 What are the…

A: Here, the partial sum of the series ∑n=1∞an is given by: SN=6-2N2 where SN=∑ann=1N

Q: Find the the maclaurin following: four Series nonzero terms of the for each

A: I am going to solve the given problem by using some simple calculus to get the required result.

Q: 9.3 For each series, is it divergent or convergent? Give reasons. In n ( 6) Σ n=1 (b) sin(n) n3/2…

A: use the comparison test to check the convergence of both series:-

Q: To find: The nth term of the power series of lnx and cosx for the power n= 5 and for a value of x =…

A:

Q: O The series Σ7-"[2 +5¹] n=1 converges to none of the choices the above converges to converges to…

A: The above series converges to

Q: 00 2n(n+ 1)x" n=1

A:

Q: 2. Consider that > (x – 2)* is a geometric series. As such, for what values of x will this infinite…

A: Given that ∑k=0∞x-2kTo find : Value of xSolution :∑k=0∞x-2kRatio test ak=x-2k…

Q: Is it true that if E=1a, is a divergent series of positive numbers, then there is also a divergent…

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Q: Euler found the sum of the p-series with p = 4: 3(4) = n- 1 06 Use Euler's result to find the sum of…

A: here the given , ζ4=∑n=1∞1n4= π490      .............................(1) now we have to find part (a…

Q: 2n Question 1: If the nth partial sum of the series a, is S, find: n+1 a) Ča, . Ъ) а,-

A: We are authorized to answer one question at a time, since you have not mentioned which question you…

Q: Suppose that an = 1, that S b, = -1, that a, = 2, and b1 = -3. Find the sum of the indicated series.…

A: given,

Q: 20. Find the sum, if it exists, of the infinite geometric series S =1+++ +.... O S= 3 OS= 2 O The…

A: Common ratio(r)  is obtained by the ratio of second term to the first term of the GP

Q: 5) /3

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Q: Ex Find the Taylar series gererated by flwat at a=2. where, if anywhere, does the to 오? series…

A: We will use the definition

Q: 1 8. Find the value of p for which the series is convergent. n In n· (In (In n))² n=3

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Q: For what series is the computation 00 00 Eax = £ (azk+azk–1) k=1 k=1 valid? Is this a rearrangement?

A: The given expression is ∑k=1∞ak=∑k=1∞a2k+a2k-1. As we know expansion of the series is:…

Q: Suppose Ea, and Eb, are series with positive terms and Eb, is known to be divergent. If a, > b, for…

A:

Q: Euler found the sum of the p-series with p = 4: 3(4) = Ë 1 90 n = 1 Use Euler's result to find the…

A:

Q: 7. (2) Prove by induction the standard formula for the sum of the first n +1 terms of a geometric…

A: To prove: ∑k=0nark = arn+1-1r-1 , r≠1Proof : we will prove by principle of mathematical inductionLet…

Q: Find a formula for the general term an of the infinite series 1/3, 1/9, 1/27, 1/81+... Assume the…

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Q: Find the the first maclaurin following: four Series nonzero for each terms of the

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Q: Let ar be the rth term of the series and the sum is defined by, п(п + 1)(п + 2)(п + 3) ar = 8…

A: Given problem is :

Q: Solve for the value of k that makes the series converge. 3" nk n=1 (Use symbolic notation and…

A:

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.