By C J Tranter

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After some point, k = M , all the uk and vk are positive and uk ≤ vk . The sum an = n M vk then forms an increasing sequence of real numbers, so by assumption this has a limit (the series converges). The sum bn = n M uk is an increasing sequence of real numbers also. Because uk ≤ vk you immediately have bn ≤ an for all n. bn ≤ an ≤ lim an n→∞ this simply says that the increasing sequence bn has an upper bound, so it has a limit and the theorem is proved. Ratio Test To apply this comparison test you need a stable of known convergent series.

918 and Stirling gives 8 051 701. 2—Infinite Series 33 Asymptotic You may have noticed the symbol that I used in Eqs. 21). “∼” doesn’t mean “approximately equal to” or “about,” because as you see here the difference between n! and the Stirling approximation grows with n. 17) describes the distribution of the results when you toss a coin. It’s straight-forward to derive this from Stirling’s formula. In fact it is just as easy to do a version of it for which the coin is biased, or more generally, for any case that one of the choices is more likely than the other.

For the same accuracy, how many terms does this improved sum take? 25 The electric potential from one point charge is kq/r . For two point charges, you add the potentials of each: kq1 /r1 + kq2 /r2 . Place a charge −q at the origin; place a charge +q at position (x, y, z ) = (0, 0, a). Write the total potential from these at an arbitrary position P with coordinates (x, y, z ). Now suppose that a is small compared to the distance of P to the origin r = x2 + y 2 + z 2 and expand your result to the first non-vanishing power of a, or really of a/r.

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Categories: Mathematical Physics