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20b(4b3)3
Absolute convergence of the series ∑ n = 1 ∞ ( − 1 ) n ln ( cos ( 1 n ) )
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Simplification of cos4(x)+sin4(x) (sinx)4+(cosx)4=(1−cos2x)24+(1+cos2x)24 =1−2cos2x+(cos2x)2+1+2cos2x+(cos2x)24 =1+(cos2x)22 its correct?
Four positive numerator-1 fractions summing to 3/7This year's seventh-grade olympiad sponsored by Tel Aviv University, round two, held a couple of days ago, had this (translated by yours truly) as its third question:Find four distinct natural numbers, a, b, c, and d, such that 1 a + 1 b + 1 c + 1 d = 3 7 My solution: 3 7 = 1 7 + . 285714 ¯ = 1 7 + 1 5 + .0 857142 ¯ = 1 7 + 1 5 + 6 70 = 1 7 + 1 5 + 1 70 + 1 14 Tweaking that a bit led me to 1 7 + 1 4 + 1 32 + 1 224 Question: Are there more solutions? Especially: Are there infinitely many?
An insect pest population doubles every 18 days. If an insecticide kills 90% of the insects, how often should it be applied to keep in insect population in check?I tried using this exponential growth/decay model: A = P ( A 1 P 1 ) t / t 1 where P 1 (initial value), A 1 (new value), t (time period).however, might this be the right manner to do it? i'm no longer quite sure how to interpret (and for this reason clear up) this problem.
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Why can we use empirical standard deviation when computing mean confidence intervalsI'm reviewing some basic statistics, and I'm asking myself questions on things I used to take for granted when I first saw them years ago. I'm going to state things as I understand them, so there might be a mistake in the following.Consider a random variable X following a distribution of mean μ and standard deviation σ. We measure n samples of X, and observe an empirical mean x ¯ and empirical std s.Because we're observing samples, x ¯ and s are random variables themselves. We should therefore not use x ¯ and μ interchangeably, and the same goes for s and σ. Instead, people compute confidence interval on μ based on x ¯ . By the central limit theorem, if n is large enough, we can say that x ¯ ∼ N ( μ , σ n ).When computing confidence intervals, we usually use σ x ¯ = s n . If all of this is correct, my question is the following: since both s and x ¯ are random variables, why does it seems to be ok to consider s = σ when computing confidence intervals for μ?
Find the surface area of the helicoid (spiral ramp) given by 𝑟⃗(𝑢, 𝑣) = 〈𝑢 cos 𝑣 , 𝑢 sin 𝑣 , 𝑣〉 for 0 ≤ 𝑢 ≤ 1 and 0 ≤ 𝑣 ≤ 𝜋. Use the following function to graph the helicoid in GeoGebra. surface(x,y,z, parameter variable 1, lower bound, upper bound, parameter variable 2, lower bound, upper bound)
Calculate. m n − n 5 m + n n
Attempt to view irrational number as a fractionI am wondering if an irrational number can be represented as a fraction in this way:For example (to represent π): π = 3.14159265359... = 314159265359... 100000000000... In the fraction 314159265359... 100000000000... , the numerator is an integer whose digits have the same order as digits of π, and the denominator is simply 10 ( # o f d i g i t s o f n u m e r a t o r − 1 ) . Isn't an irrational number represented as a fraction in this way? Probably I misunderstand the concept of the irrational number. Thanks in advance.
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