series test cheat sheet, Cheat Sheet of Mathematics

Download series test cheat sheet and more Cheat Sheet Mathematics in PDF only on Docsity! Strategy for Testing Series We now have several ways of testing a series for convergence or divergence; the problem is to decide which test to use on which series. In this respect testing series is similar to inte- grating functions. Again there are no hard and fast rules about which test to apply to a given series, but you may find the following advice of some use. It is not wise to apply a list of the tests in a specific order until one finally works. That would be a waste of time and effort. Instead, as with integration, the main strategy is to classify the series according to its form. 1. If the series is of the form , it is a -series, which we know to be conver- gent if and divergent if . 2. If the series has the form or , it is a geometric series, which con- verges if and diverges if . Some preliminary algebraic manipula- tion may be required to bring the series into this form. 3. If the series has a form that is similar to a -series or a geometric series, then one of the comparison tests should be considered. In particular, if is a rational function or algebraic function of (involving roots of polynomials), then the series should be compared with a -series. (The value of should be chosen as in Section 8.3 by keeping only the highest powers of in the numerator and denominator.) The comparison tests apply only to series with positive terms, but if has some negative terms, then we can apply the Comparison Test to and test for absolute convergence. 4. If you can see at a glance that , then the Test for Divergence should be used. 5. If the series is of the form or , then the Alternating Series Test is an obvious possibility. 6. Series that involve factorials or other products (including a constant raised to the th power) are often conveniently tested using the Ratio Test. Bear in mind that as for all -series and therefore all rational or algebraic functions of . Thus, the Ratio Test should not be used for such series. 7. If , where is easily evaluated, then the Integral Test is effec- tive (assuming the hypotheses of this test are satisfied). In the following examples we don’t work out all the details but simply indicate which tests should be used. EXAMPLE 1 Since as , we should use the Test for Divergence. EXAMPLE 2 Since is an algebraic function of , we compare the given series with a -series. The comparison series for the Limit Comparison Test is , where bn
sn 3 3n 3
n 3
2 3n 3
1 3n 3
2  bn pnan 
n
1 sn 3  1 3n 3  4n 2  2 n l
an l 1 2  0 
n
1 n  1 2n  1 x
1 f x dxan
f n n pn l
 an1
an l 1 n  1nbn 1n1bn lim n l
an  0   an   an n pp n an p  r   1 r   1  arn arn1 p  1p  1 p 1
np 1 EXAMPLE 3 Since the integral is easily evaluated, we use the Integral Test. The Ratio Test also works. EXAMPLE 4 Since the series is alternating, we use the Alternating Series Test. EXAMPLE 5 Since the series involves , we use the Ratio Test. EXAMPLE 6 Since the series is closely related to the geometric series , we use the Comparison Test.  1
3n 
n
1 1 2  3n k! 
k
1 2k k! 
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1 1n n 3 n 4  1 x
1 xe x2 dx 
n
1 nen 2 2 ■ STRATEGY FOR T ES T ING SER I ES Exercises 1–34 Test the series for convergence or divergence. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 
n
1 n 2  1 n 3  1 
n
0 n! 2
5
8
  
3n  2 
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1 sin n
n
1 3nn 2 n! 
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1 1n n n 2  25
n
2 1n1 n ln n 
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1 n 2en 3
k
1 k 2ek 
k
1 2 k k! k  2!
n
2 1 nsln n 
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1 1 n  n cos2n
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1 3n1 23n 
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1 1n1 n  1 n 2  n
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1 1 n 2  n 
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1 n  1 n 2  n
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1 n 2  1 n 2  n 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 
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1 (sn 2  1)
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1 sin1
n sn 
n
2 1 ln nln n
k
1 5 k 3 k  4 k 
j
1 1 j sj j  5
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1 tan1n nsn 
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1 e 1
n n 2
k
1 k ln k k  13 
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1 n 2  1 5n
n
1 n! e n 2 
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1 cosn
2 n 2  4n
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1 tan1
n 
n
1 sn 2  1 n 3  2n 2  5
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1 22n n n 
k
1 k  5 5k
n
1 1n ln n sn 
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2 1n1 sn  1
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1 1n21
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