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THE INTEGRAL TEST AND ESTIMATES OF SUMS
Example: Test the series for convergence or divergence.
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THE INTEGRAL TEST AND ESTIMATES OF SUMS
THEOREM: (Integral Test) a continuous, positive, decreasing function on [1, inf) Convergent Convergent Divergent Dinvergent Example: Remark: Test the series for convergence or divergence. sequence of positive terms.
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THE INTEGRAL TEST AND ESTIMATES OF SUMS
THEOREM: (Integral Test) a continuous, positive, decreasing function on [1, inf) Convergent Convergent Divergent Dinvergent Example: Test the series for convergence or divergence.
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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
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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.
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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?
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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?
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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?
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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.
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THE INTEGRAL TEST AND ESTIMATES OF SUMS
FINAL-081
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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??
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THE INTEGRAL TEST AND ESTIMATES OF SUMS
Bounds for the Remainder in the Integral Test Convergent by integral test Error (how good) good approximation Example:
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THE INTEGRAL TEST AND ESTIMATES OF SUMS
REMARK: We can estimate the sum ESTIMATING THE SUM OF A SERIES 1 2 3 4 5 10 20 40 50 1000 11000 21000 61000 71000 81000 91000
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THE INTEGRAL TEST AND ESTIMATES OF SUMS
REMARK: We can estimate the sum ESTIMATING THE SUM OF A SERIES Example: Estimate the sum How accurate is this estimation?
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THE INTEGRAL TEST AND ESTIMATES OF SUMS
Example: Estimate the sum How accurate is this estimation?
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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.
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THE INTEGRAL TEST AND ESTIMATES OF SUMS
TERM-112
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THE INTEGRAL TEST AND ESTIMATES OF SUMS
TERM-102
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