Enter question here... Let (xn)n N be a sequence in R. If xn Solution If none of
ID: 3087412 • Letter: E
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Let (xn)n N be a sequence in R. If xnExplanation / Answer
If none of the above works. The following tests should be used, if one is inconclusive or irrelevant try another. It is not always obvious which one to try first, but with practise you may get better at deciding, but there is no set method as which to choose. The Comparison test. Assume you have two positive term series, a(n) and b(n). Then i) If the infinite sum of b(n) converges and a(n) is less than b(n) (for sufficiently large n) then the sum of a(n) also converges. ii) If b(n) diverges and a(n)>b(n) then a(n) also diverges. Example, say we have the series 2/x; we can compare this to 1/x. Because we already now that 1/x is divergent, and because 2/x > 1/x, then it follows that 2/x also diverges. So, the basic method is to use a known series to determine whether the unknown series converges or diverges. The Limit Comparison Test. If a(n) and b(n) are positive term series and if the limit of a(n)/b(n) exists and is greater than 0 then both series converge or both diverge. Again, this requires using a known series, the method is generally to pick a second series whose highest power is the same as the highest power of the given one. For example if you were given 1/(x^3+2x+1), then it makes sense to compare it to 1/(x^3). The Integral Test. If a function is positive valued, continuous and decreasing for x greater than or equal to one. Then the infinite series f(n) converges if: the integral between 1 and infinity of f(x) exists, and diverges if the integral doesn't exist. So basically, integrate your function and find the limit to infinity. if it exists then the series converges, if it doesn't then the series diverges. The Alternating Series Test. If a(k)>a(k+1)>0 for sufficiently large k, and the limit of a(n) exists then the alternating series (-1)^n a(n) converges. Put more simply, if you have an alternating series, a series where each term changes sign. Then eliminate the alternating part of the function and find the limit of what's left, if the limit exists then the series converges. The Ratio Test. Given an infinite series a(n), you should find a(n+1), the general term for the next term in the series. Then compute a(n+1)/a(n), take a modulus if necessary. Find the limit of this, if the limit exists it tells you one of three things. 1) If the limit is less than one then the series converges. 2) If the Limit is greater than one the series diverges. 3) If the limit equals one then the test is inconclusive. These are the main convergence tests, and are extremely useful. If none of these work, it is most likely that the problem is unsolvable or you've made a mistake. They can be extended to more things such as Power Series, Taylor Series and much more, it is very useful to understand these tests as there really is no other simple way of determining convergence.
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