Calc1231W11T1 - Integral test for convergence

Sunday, 14 October 2018

3:51 PM

The integral test links the convergence of a sum to the convergence of an integral Let be an infinite series of positive terms. Suppose that f(x) is a continuous; positive and decreasing function where There are two cases to consider: ak is divergent i) If f(x) dx is divergent; then O f(k) = ak fork = 1, 2, 3, This is shown in the first figure because the area of the rectangles is greater than the 1 under the curve ii) If dc is convergent, then 1 under the curve (11 convergent 02 2 03 3 4 5 O infinite finite O area O area This is shown in the second figure because the area of the rectangles is less than the a2 2 03 3 4 a5 5 ak is convergent But this is equivalent to the statement that ak is convergent because the difference between these two sums is Actually, the second diagram only shows that finite

 

We can use the integral test to determine if the series 3 converges. First we take a continuous, positive and decreasing function f : R —+ R where f(n) Now for any positive N dc f@) dc 3 for all n Such a function is 1 Then dc 1 So the series (1) is Note: the Maple syntax for oo is infinity. lim 1/320 1

 

We can use the integral test to determine if the series converges. First we take a continuous, positive and decreasing function f : R —+ R where f(n) xlsqrt(xA2+1) for all n Such a function is Now for any positive N dc 1 -sqrt(2)+sqrt(W2+1) Then dc 1 So the series (1) is Note: the Maple syntax for oo is infinity. lim dc 1 infinity

 

An important application of the integral test is to p-series. Recall that the sum converges precisely when p > 1 We can check this fact by defining f@) continuous negative discontinuous positive Increasing decreasing Now, if p 1 then 1 n=l for x > 1 _ This function is (check all that apply): 1 Hence the convergence or divergence of the integral depends on p. • Ifp< 1 then I • Ifp> 1 then I infinity l/(p-l) 1 1 O and the p-series and the p-series x dc diverges converges lim dc 1 O O by the integral test by the integral test • We analyse the remaining case p = 1 in the next question. Note: the Maple syntax for oo is infinity.

 

Now we consider the case of a p-series with p converges precisely when converges. By definition and so Hence the sum diverges by the integral test. 1 According to the integral test 1 dc 1 1 — dc 1 lim 1 Ill(N) 1 — dc 1 — dc infinity 1

 

This is the famous harmonic series, whose divergence was first discovered by Nicole Oresme (1323-1382) His argument was that 3 4 and since this series diverges, so does the harmonic series. 2 5 8 6 8 7 8 8 8

 

 

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