Convergence Tests Academic Resource Center Series • Given a sequence {a0, a1, a2,…, an } • The sum of the series, Sn= n a k k 1 • A series is convergent if, as n gets larger and larger, Sn goes to some finite number. • If S does not converge, and S goes to ∞, then the series is n n said to be divergent Geometric and P-Series • The two series that are the easiest to test are geometric series and p-series. • Geometric is generally in the form ar k • P-series is generally in the form k 1 1 p n 1 n Geometric Series • A geometric series is a series in which there is a constant ratio between successive terms • 1 +2 + 4 + 8 + … each successive term is the previous term multiplied by 2 • each successive term is the previous term squared. 1 1 1 1 ... 2 4 16 256 Geometric Series • Sn= = ar +ar^2 + ar^3 + … +ar^k • As a result, if |r|<1, the geometric series will converge to , and if |r| 1 the series will diverge. a ar k k 1 1 r P-Series • Given a series • This series is said to be convergent if p>1, • And divergent if p 1 1 p n 1 n Geometric and P-Series Examples n n 1 This series is 3 3 The Sum of the series, S 3 geometric with a=3 4 n 1 4 and r =3/4. Since a n 1 n 1 r<1, this series will S converge. So S= 3/(1-3/4) = 12 1 r 1 Here, p=3, so p>1. Therefore our series 3 will converge n 1 n 1 1 1 2 Here, p=1/2, so p<1. Therefore our series will diverge n n n 1 n 1 Convergence Tests • Divergence test • Comparison Test • Limit Comparison Test • Ratio Test • Root Test • Integral Test • Alternating Series Test Divergence Test • Say you have some series a n • The easiest way to see if a series divergesn 0 is this test • Evaluate L= Lim • If L 0, the series diverges • If L=0, then this test is inconclusive a n n Divergence Test Example n 2 Let’s look at the limit 2 n 1 5 n 4 of the series n 2 n 2 1 Lim 2 Lim 2 0 Therefore, this series is divergent n 5 n 4 n 5 n 5 1 n 2 The limit here is equal to zero, so this test is inconclusive. n 1 However, we should see that 1 this a p-series with p>1, therefore this will converge. Lim 2 0 n n Comparison Test • Often easiest to compare geometric and p series. • Let and be series with non-negative terms. • If for k, as k gets big, then ak bk • If converges, then converges • If diverges, then diverges b a k k Comparison Test Example 1 Test to see if this series converges n using the comparison test n 1 3 1 1 which is a geometric series so This is very similar to n it will converge n 1 3 1 1 And since our original series will also converge 3 n 3 n 1 n 1 n 1 Limit Comparison Test • Let and be series with non-negative terms. • Evaluate Lim • If lim=L, some finite number,a then both and either converge or diverge.k ak bk k b k • and are generally geometric series or p-series, so seeing whether these series are convergent is fast. Limit Comparison Test Example 9 n Determine whether this series n n n 1 3 10 converges or not 9 n 9 n 9 3 10 n Lim n 1 0 Compare it with so n n 9 n 1 10 n 1 10 10 n 9 And since Is a geometric series with r<1, this series converges, n 1 10 therefore so does our original series Ratio Test • Let be a series with non-negative terms. • Evaluate L= Lim a k 1 • If L < 1,a then converges k k a • If L > 1, then diverges k • If L = 1, then this test is inconclusive Ratio Test Example Test for convergence 3 a n n Look at the limit of n 1 1 n a n n 1 3 (1) n 1 (n 1) 3 3 n 3 n 1 (n 1) 3 Lim n 3 Lim n 1 3 n (1) n n 3 n Since L<1, this series 3 n will converge based on the ratio test 1 n 1 1 1 1 Lim ( ) 3 Lim (1 ) 3 1 3 n n 3 n n 3 Root Test • Let be a series with non-negative terms. an • Useful if involves nth powers • Evaluate L= Lim 1 • If L < 1, is convergent n (a n ) • If L > 1, is divergent a n an • If L = 1, then this ntest is inconclusive a n Root Test Example Test for convergence 4 n 5 n Lets evaluate the limit, L =Lim ( ) 1 5 n 6 n n 1 (a n ) 1 4 n n 5 4 n 5 4 Lim (( ) n ) n Lim 1 n n 5n 6 5 n 6 5 By the root test, since L<1, our series will converge. Integral Test • Given the series , let = f(k) a k • f must be continuous, positive, and decreasing for x > 0 • will converge only if converges. • If a diverges, then the series will also diverge. k • In ageneral, however, k f ( x)dx k 0 0 a f ( x )dx k 1 k 1 Integral Test Example Test for convergence Since x>0, f(x) is 1 1 continuous and So let positive. 3 f ( x ) (2 n 1) (2 x 1) 3 f’(x) is negative so n 1 we know f(x) is decreasing. Now let’s look 1 1 1 t 3 dx 2 Lim [ 2 ] at the integral 1 (2 x 1) 2 t (2 x 1) 1 1 1 1 Lim ( 2 2 ) t (2 t 1) (3) 9 Since the integral converged to a finite number, our original series will also converge Note: Series will most likely not converge to 1/9, but it will converge nonetheless. Alternating Series Test • Given a series (1) k a , where is positive for all k k a k • IF • for all k, and • Lim = 0 k a a Then kthe1 seriesk is convergent Alternating Series Test Example Test for convergence 2 n 1 n Check: 1 3 Is this series decrease- yes n 1 n 4 Is the Lim=0? n 2 Lim 3 0 Yes n n 4 Therefore, , is convergent. More Examples cos n 1 n n 2 1. 2. 3 4 2 6 n 1 n n 1 1 n n n 10 1 4. 3. 2 n 1 n 2 n ln n n 1 (n 1)4 Answers • 1. By Alternating series test, series will converge • 2. By the comparison test, series will diverge • 3. By the ratio test, series will converge • 4. By the integral test, series will diverge .