Recent content by Simkate

Let f be a continuous on the closed and bounded interval [a,b] and x_1, x_2, …, x_n ∈ [a,b]. Show that there necessarily exists ξ ∈ [a,b] such that: f (ξ= [f(x_1) + f(x_2) + …f(x_n)] / n How can I start this problem i am really confused! please help !

I just want to know if what i did is correct...please help Thank YOU! ∑(n =200 to ∞) ( -1)^n (1-1/n)^n = (-1 + 1/n) This is a alternation Series therefore the 2 condtions need to be satisfied for it to be Convergent. i) is true ii) lim b_n = 0 ? L= lim (n--> ∞) (1-1/n)^n =...

:) thank u so much i really appreciate it...it was so helpful

Thank You i did see my silly mistake their and i have corrected now it makes sense. so at the end the lim (n--∞) 2n+1/n+1 * [ n/n+1]^n i can divide all the terms by n? to make it [ (2+1/n)/ (1+1/n) ] * [(1)/(1+1/n)^n] = 2/e < 1 therefore the series is absoluetely convergent?

What signaficance does the n=10 have? the limit is still n to infinity right? Thank you for that correction: Now i have got: after cancelling out terms through the ratio test i ended up with -lim(n-->∞) [(2n+2) (2n)^n] / [(n+1) (2n+2)^n] -lim(n-->∞) [2n+2/ n+1] * [...

= (2n/2n+1)^n so it is conditionally convergent?

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Given the following two skew lines: L1: (0, 4, -3) + s(-1, 1, 3) L2: (1, 2, 5) + t(-3, 2, 5) Find the shortest distance. MY WORK:: Cross-product of the lines (-1, 1, 3) X (-3, 2, 5) = (-1, -4, 1) with length 3*sqrt(2) Vector between the points (0, 4, -3) - (1, 2, 5) = (-1, 2, -8)...

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Find the supremum and infimum of S, where S is the set S = {√n − [√n] : n belongs to N} . Justify your claims. (Recall that if x belongs to R, then [x] := n where n is the largest integer less than or equal to x. For example, [7.6] = 7 and [8] = 8) ----I found my infimum to be 0 and...

For the following power series, find ∑ (4^n x^n)/([log(n+1)]^(n) (a) the radius of convergence (b) the interval of convergence, discussing the endpoint convergence when the radius of convergence is finite. -------------------------------------------------------------------------------...

Show that for any two integers a, b , (a+b)^2 ≡ a^2 + b^2 (mod 2) I have my solution below i wanted someone to help chekc if i have done anything wrong. Thank You for your help. The thing that is going on here is that 2x = 0 (mod 2) for any x. If x = ab, then 2ab = 0 (mod 2). We see...

so do i jus use the p-test don't i have to use the integral test making u=ln (n+1) ?

Does the following series converge(absolutely or conditionally) or diverge? E= (-1)^(n)sin (1/n)/(ln(1+n))^(2) can anyone help me solve this or atleast tell me which series test to use? thank u

The sequence b(n)= npicos(npi) coverges or diverges? if it converges what is the limit? My Work When n is even cosnpi= o and b(n)=o (therefore when n is oven it converges) but when n is odd cosnpi=-1 and b(n)= - infinity( thereofre when n is even it diverges) is this correct?