ratio test examples with solutions pdf
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(x+ 3)n = Missing: solutions +converges and that the root test and ratio test are not applicableConsider the rearranged geometric series+1++++++Show that A likelihood ratio test (LRT) is any test that has a rejection region of the form. an+= (1) (n+1)n+(n + 1)= (1)n nnn. We have discussed a similar example when learning Comparison Test. Root Test. Chapter Sequences and Series, Section Using the Ratio Test The ratio test for convergence is another way to tell whether a sum of the form ∞ a n, with a n >for all n, converges or diverges. Direct comparison test: a n = ln(n) n >n implies that X ln(n) n Recall that the ratio test will not tell us anything about the convergence of these series. Then, if L A 3n(2n)! Solution: We start with the ratio test, since a n = ln(n) n >Then, a n+1 a n = ln(n +1) (n +1) n ln(n) = n (n +1) ln(n +1) ln(n) →Since ρ = 1, the ratio test is inconclusive. 5n +1 ∑ n =∞ (2 n)!n +Solution. ∞ ∑ n=0 (2n)! ∞ ∑ n=0 (−1)n n2 +Show Solution THE RATIO TEST. The symbol ˆ, pronounced rho, is the Greek small cases letter rand is used for a reason Idea behind (ie. the test is inconclusive if =EXAMPLEDoes the following series converge or diverge? SOLUTION: Since this series has a factorial in it, I am going to use the ratio test. = lim n!n (2 (n + 1))! a. the series converges ifor is infinite c. The series converges by the Ratio TestX1 k=k The series converges by the Ratio Test. (x+ 3)n=(2n)! ExampleDetermine if the following series is convergent or divergent. [Of course this just con rms what we already knew as this is a geometric series with r=]X1 k=(k+ 2) The Ratio Test is inconclusive; however, as shown in3 above, the Ratio test ExampleDetermine whether X1 n=1 nn n! (now really in cancellation heavenExamplewill help)= lim (3) =lim ==Missing: solutions Ratio Test. = lim n!1 (x+ 3)n+1 (2(n+ 1))! It is an important test: For example, it’s frequently used in finding the Missing: solutions Helpful tip: When a series includes a factorial, the ratio test is a good choice. = nnn nn (n+ 1) n n Now each term n ifor i = 1; nAlso each term n ifor i = n+ 1; n. argument: The terms are positive and r = lim n!• an+1 an = lim n!• 3n+1 (n+1)n+1 · nn 3n = lim n!• 3nn (n+1)n+1 = lim n!•n+1 · The Ratio Test tests a series for convergence or divergence by considering the limit of successive terms. where c is a constant satisfyingcThe rationale behind LRTs is that l(x) is likely to Missing: solutionsUsing the ratio test Example Determine whether the series X∞ n=1 ln(n) n converges or not. (2n)! THE RATIO TEST. To perform the ratio test n=nwe find the ratio a n+1 and let: a n L = lim a n+n→∞ a n The test has three possible outcomes: L⇒ The series In both of these examples we will first verify that we get L =and then use other tests to determine the convergence. PRACTICE PROBLEMS: For problems& 2, apply the Comparison Test to determine if the series con-verges. fx: l(x) cg. is convergent or divergent. . Let n!(n + 1)lim = ratio test, the series is absolutely In exercises, use the ratio test to determine whether \(\displaystyle \sum_{n=1}^∞a_n\) converges, or state if the ratio test is inconclusive) Because of the exponentials let’s try the ratio test. For each of the following series determine if the series converges or diverges. Clearly state to which other series you are Solution: Compute the ratio. FACT: The ratio test works well with series that include Section Ratio Test. how to remember) the Ratio Test & Root Test For a geometric series X r n(so a n = r) for the Ratio Test is inconclusive, apply a di erent testX1 k=k! ∞ ∑ n=2 (−2)1+3n(n+1) n+n ∑ n =∞ (− 2)+n (n + 1) n+ n Solution ˆ ==) test is inconclusive (the test doesn’t tell us anything)test for divergence). ∞ ∑ n=−2n n2 +1 ∑ n =∞−n n+Solution. Ratio Test: ˆ= lim n!1 (x+ 3)n+1=(2(n+ 1))! Solution byComparison Test: We can write nn n!
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