Calculate the Bernoulli numbers B8, B10, and B12 . The...

Calculate the Bernoulli numbers $B_{8}, B_{10}$, and $B_{12} .$ The sequence of Bernoulli numbers is usually completed by setting $B_{0}=1, B_{1}=-\frac{1}{2}$, and $B_{k}=0$ for $k$ odd and greater than 1 .

Key Concepts

Bernoulli Numbers

Bernoulli numbers are a sequence of rational numbers that arise in the expansion of the generating function x/(e^x - 1) and play a significant role in number theory, analysis, and combinatorics. They are used in various formulas, including the Euler-Maclaurin formula and expressions for sums of powers of consecutive integers.

Generating Function

The generating function for Bernoulli numbers, expressed as x/(e^x - 1) = ? (B_n * x^n/n!), is a central tool for defining and exploring their properties. This power series representation compactly encodes the entire sequence and is instrumental in deriving identities and recurrence relations for Bernoulli numbers.

Even-Odd Index Property

A noteworthy property of Bernoulli numbers is that, except for B_1, all Bernoulli numbers with odd indices greater than one are zero. This characteristic simplifies many calculations and theoretical derivations by reducing the number of nonzero terms in formulas involving Bernoulli numbers.

Recurrence Relations

Bernoulli numbers can also be computed using recurrence relations that express each number in the sequence in terms of the previous ones. These recursive formulas provide an alternative method for obtaining Bernoulli numbers and are especially useful in proofs and computational algorithms in number theory and other fields.

Transcript

00:02 All right, so this question asks us some questions about a binomial distribution with 10 trials, each having a success rate of 0 .1.

00:12 So, part a asks for probability that we get zero successes out of 10.

00:21 So how can that happen? well, there's only one way that can happen, and that is if you get a failure.

00:31 10 times in a row, which is just 0 .9 to the 10th power, which works out to be 0 .349.

00:42 All right.

00:46 Part b asks, what is the probability that you get exactly two successes? and that we can plug into the binomial distribution formula.

00:57 So that is 10 choose 2 times probability of success times to the power of two successes times probability of failure to the number of failures and that works out to be 0 .194 which alternatively, if you have a ti84, that's binomial pdf of 10 .1 and 2, which will give the same answer.

01:43 All right.

01:47 Part c.

01:49 It wants probability that x, that x, is less than or equal to 2, which that's equal to probability that x equals 0 plus probability x equals 1 plus probability that x equals 2.

02:16 Which you can do this by plugging each of these numbers into the binomial formula like we did for part b.

02:24 Or if you have a t -i -84, you can just do binomial cdf, c because it's cumulative, 10 trials, 0 .1 probability of success, and we want to sum all the probabilities up to 2.

02:44 And that works out to be 0 .930...

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