THE INTEGRAL TEST AND ESTIMATES OF SUMS Example: Test the series for convergence or divergence.

THE INTEGRAL TEST AND ESTIMATES OF SUMS THEOREM: (Integral Test) Remark: a continuous, positive, decreasing function on [1, inf) Convergent Convergent sequence of positive terms. Divergent Dinvergent Example: Solution: Test the series for convergence or divergence. Since this improper integral is convergent, the series is also convergent by the integral test.

THE INTEGRAL TEST AND ESTIMATES OF SUMS THEOREM: (Integral Test) a continuous, positive, decreasing function on [1, inf) Convergent Convergent Divergent Dinvergent Example: Solution: Test the series for convergence or divergence. Since this improper integral is divergent, the series is also divergent by the integral test.

THE INTEGRAL TEST AND ESTIMATES OF SUMS THEOREM: (Integral Test) a continuous, positive, decreasing function on [1, inf) Convergent Convergent Divergent Dinvergent REMARK: When we use the Integral Test, it is not necessary to start the series or the integral at n = 1 . For instance, in testing the series

THE INTEGRAL TEST AND ESTIMATES OF SUMS THEOREM: (Integral Test) a continuous, positive, decreasing function on [1, inf) Convergent Convergent Divergent Dinvergent REMARK: REMARK: When we use the Integral Test, it is not necessary to start the series or the integral at n = 1 . For instance, in testing the series Also, it is not necessary that f(x) be always decreasing. What is important is that f(x) be ultimately decreasing, that is, decreasing for larger than some number N.

THE INTEGRAL TEST AND ESTIMATES OF SUMS THEOREM: (Integral Test) a continuous, positive, decreasing function on [1, inf) Convergent Convergent Divergent Dinvergent Special Series: Example: Geometric Series Harmonic Series Telescoping Series p-series Alternatingp-series Harmonic Series is the series convergent?

Memorize: THE INTEGRAL TEST AND ESTIMATES OF SUMS Example: Special Series: Geometric Series Harmonic Series Telescoping Series p-series Alternatingp-series Example: For what values of p is the series convergent?

THE INTEGRAL TEST AND ESTIMATES OF SUMS THEOREM: (Integral Test) a continuous, positive, decreasing function on [1, inf) Convergent Convergent Divergent Dinvergent Example: P Series: For what values of p is the series convergent?

THE INTEGRAL TEST AND ESTIMATES OF SUMS Example: P Series: For what values of p is the series convergent? Example: Example: Test the series for convergence or divergence. Test the series for convergence or divergence.

THE INTEGRAL TEST AND ESTIMATES OF SUMS FINAL-081

THE INTEGRAL TEST AND ESTIMATES OF SUMS THEOREM: (Integral Test) a continuous, positive, decreasing function on [1, inf) Convergent Convergent Divergent Dinvergent REMARK: REMARK: We should not infer from the Integral Test that the sum of the series is equal to the value of the integral. In fact, Integral Test just test if convergent or divergent. But if it is convergent what is the sum??

THE INTEGRAL TEST AND ESTIMATES OF SUMS TERM-102

ESTIMATES OF SUMS This will be joined with alternating test 11.5

S = Sn + Rn S ~ Sn ~ THE INTEGRAL TEST AND ESTIMATES OF SUMS SYLLABUS: Students should know the “Remainder Estimate for the Integral Test”. Example 5a and Example 6 are excluded. ESTIMATING THE SUM OF A SERIES S = Sn + Rn S ~ Sn ~ for sufficient large n Example: Estimate the sum How accurate is this estimation?

THE INTEGRAL TEST AND ESTIMATES OF SUMS Bounds for the Remainder in the Integral Test Convergent by integral test Error (how good) good approximation REMARK: We can estimate the sum Example:

THE INTEGRAL TEST AND ESTIMATES OF SUMS REMARK: We can estimate the sum ESTIMATING THE SUM OF A SERIES 1 1.000000000000000 2 1.250000000000000 3 1.361111111111111 4 1.423611111111111 5 1.463611111111111 10 1.549767731166541 20 1.596163243913023 40 1.620243963006935 50 1.625132733621529 1000 1.643934566681562 11000 1.644843161889427 21000 1.644886448934383 61000 1.644901809303995 71000 1.644919982440396 81000 1.644921721245446 91000 1.644923077897639

THE INTEGRAL TEST AND ESTIMATES OF SUMS Example: Estimate the sum How accurate is this estimation?

15 22 Sec 11.3: THE INTEGRAL TEST AND ESTIMATES OF SUMS Facts about: (Harmonic Seris) 1)The harmonic series diverges, but very slowly. 15 the sum of the first million terms is less than 22 the sum of the first billion terms is less than 2) If we delete from the harmonic series all terms having the digit 9 in the denominator. The resulting series is convergent.

THE INTEGRAL TEST AND ESTIMATES OF SUMS TERM-112