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Princeton Review AP Calculus BC, Chapter 12: Infinite Sequences and Series

Sequences & Series

  • Sequence: An infinite succession of numbers that follow a pattern

    • Usually denoted with a subscript using aₙ

    • Usually the terms are generated by a formula, ex. aₙ=(n-1)/n beginning with n=1 is 0, 1/2, 2/3, etc.

      • n is always an integer

  • Officially:

    • “A sequence has a limit L if for any ε>0 there is an associated positive integer N such that |aₙ - L| < ε for all nN. If so, the sequence converges to L and we write lim (n → ∞) aₙ = L”

    • If a sequence has no finite limit, it diverges.

  • Infinite Series

  • An infinite series is an expression of the form a₁ + a₂ + a₃… + aₖ + …. The numbers a₁, a₂, a₃, etc. are the terms of the series

    • In other words, an infinite series is a sequence of terms where the terms are added up, there is a pattern to the order of the terms, and there are an infinite number of terms.

  • As n increases, the partial sum includes more and more terms of the series. So if aₙ approaches a limit as n approaches ∞, then this limit is likely to be the sum of all of the terms in the series, thus a convergence.

    • If the sequence of partial sums converges to a limit S, then the series is said to converge and S is called the sum of the series.

    • If the sequence of partial sums diverges (doesn’t have a limit), then the series is said to diverge which means that a divergent series has no sum.

Geometric Series

  • Used in the form of ∞Σ(n=1) a*r^(n-1)

    • These show up often on the AP exam: often asked to determine whether the series converges or diverges, and if it converges, to what limit.

    • If |r|<1, the series converges

    • If |r|>1, the series diverges

  • For example, the series (1/2)+(1/2^2)+(1/2^3) converges, while 2+2^2+2^3 diverges.

  • If a geometric series converges, you can figure its sum out by doing the following:

    • Multiply the expression by r

      • aₙ = a + ar + ar^2 + ar^3… + ar^(n-1) → raₙ = a + ar + ar^2 + ar^3… + ar^(n-1)

    • Subtract the second expression from the first

      • aₙ - raₙ = a - ar^n

    • Factor out aₙ from the left side

      • aₙ(1-r)=a-ar^n

    • Divide through by (1-r)

      • aₙ=(a(1-r^n))/(1-r)

  • If we want to find the sum of the first n terms of a geometric series, we use the following formula:

  • For an infinite geometric series, if it converges, lim (n→∞) r^n = 0, and its sum is found this way:

    • S = (a)/(1-r)

  • The nth Term Test For Divergence

  • The nth term test means we find the limit of the series and see if we get zero or not

    • If the limit = L (any number other than 0), the series diverges

    • If the limit = 0, we need to choose a different test

  • Integral Test for Convergence

    • If f is positive, continuous, and decreasing for x≥1, and aₙ=f(n), then

Either both converge or both diverge.

  • In other words, if you want to test convergence for a series with positive terms, evaluate the integral and see if it converges.

Harmonic Series & P-series

Harmonic Series: (1/n) pattern

  • Might look as though it is converging, but it diverges

  • The sums are not approaching a limit, so as the sequence of partial sus diverges, so does the sequence of partial sums for the harmonic series

p-Series

  • A series of the form (1/n^p) converges if p>1 and diverges if 0<p<1

Tests for Convergence

Comparison Tests for Convergence

  • Let 0 ≤ aₙ ≤ bₙ for all n

    • If the summation of bₙ from 1 to ∞ converges, then the summation of aₙ from 1 to ∞ converges

    • If the summation of aₙ from 1 to ∞ diverges, then the summation of bₙ from 1 to ∞ diverges

  • Essentially, if all of the terms of a series are less than those of a convergent series, then it too converges. If all of the terms of a series are greater than those of a divergent series, then it too diverges

Limit Comparison Test

Alternating Series Test for Convergence

  • Used for terms that alternate between positive and negative

  • The series using (-1)^(n+1) bₙ converges if the following conditions are all satisfied

    • bₙ > 0 (which means that the terms must alternate in sign)

    • bₙ > bₙ₊₁ for all n

    • bₙ → 0 as n → ∞

Absolute Convergence Theorem:

  • The series Σaₙ converges absolutely if the corresponding series Σ|aₙ| converges

Ratio Test for Convergence

Determining Absolute or Conditional Convergence

  • Sometimes, an alternating series will converge, but when we look at the absolute value of the terms, it doesn’t.

  • An alternating series that is not absolutely convergent is called conditionally convergent

Alternating Series Error Bound

  • When summing an alternating series with a finite number of terms, the sum will not be the same as the sum with an infinite number

  • If we want to find the error, we need to evaluate | S-Sₙ |

  • Can put a bound on the error: a number that the error is less than, which is simply the absolute value of the next term in the alternating series

    • If summing the first 10 terms in an alternating series, the error will be less than the absolute value of the eleventh term

Finding Taylor Polynomial Approximations of Functions

  • We can approximate a function on an interval centered at x=a by using a Taylor series (if centered about x=0, it’s a Maclurin series).

  • In a Taylor polynomial, the terms are found from thee derivatives of f as follows:

    • If f can be differentiated n times at a, then the nth Taylor polynomial about x=a is

    • pₙx = f(a) + f’(a)(x-a) + (f’’(a)/2!)(x-a)^2+ (f’’’(a)/3!)(x-a)^3+… + (f^(n)(a)/n!)(x-a)^n(x-a)^n

Lagrange Error Bound

  • The “error bound” of a Taylor polynomial, aka the Lagrange error

  • If you are finding an nth degree Taylor polynomial, a good approximation to the error bound is the next nonzero term in a decreasing series.

Radius and Interval of Convergence of Power Series

  • Going from n = 0 to n = ∞ instead of n = 1 to n = ∞

  • Radius of convergence: R.

    • If the series converges only for x=0, the radius of convergence is 0

    • If the series converges absolutely for all x, the radius is all x

    • Set of all values of X (-R, R) is the interval of convergence

Representing Functions as Power Series

  • We can use differentiation and integration of power series to find the power series of other functions

  • Can approximate the integral using its Maclurin and approximate the solution using the aforementioned series

  • Find a new Taylor series from a known Taylor series.

Princeton Review AP Calculus BC, Chapter 12: Infinite Sequences and Series

Sequences & Series

  • Sequence: An infinite succession of numbers that follow a pattern

    • Usually denoted with a subscript using aₙ

    • Usually the terms are generated by a formula, ex. aₙ=(n-1)/n beginning with n=1 is 0, 1/2, 2/3, etc.

      • n is always an integer

  • Officially:

    • “A sequence has a limit L if for any ε>0 there is an associated positive integer N such that |aₙ - L| < ε for all nN. If so, the sequence converges to L and we write lim (n → ∞) aₙ = L”

    • If a sequence has no finite limit, it diverges.

  • Infinite Series

  • An infinite series is an expression of the form a₁ + a₂ + a₃… + aₖ + …. The numbers a₁, a₂, a₃, etc. are the terms of the series

    • In other words, an infinite series is a sequence of terms where the terms are added up, there is a pattern to the order of the terms, and there are an infinite number of terms.

  • As n increases, the partial sum includes more and more terms of the series. So if aₙ approaches a limit as n approaches ∞, then this limit is likely to be the sum of all of the terms in the series, thus a convergence.

    • If the sequence of partial sums converges to a limit S, then the series is said to converge and S is called the sum of the series.

    • If the sequence of partial sums diverges (doesn’t have a limit), then the series is said to diverge which means that a divergent series has no sum.

Geometric Series

  • Used in the form of ∞Σ(n=1) a*r^(n-1)

    • These show up often on the AP exam: often asked to determine whether the series converges or diverges, and if it converges, to what limit.

    • If |r|<1, the series converges

    • If |r|>1, the series diverges

  • For example, the series (1/2)+(1/2^2)+(1/2^3) converges, while 2+2^2+2^3 diverges.

  • If a geometric series converges, you can figure its sum out by doing the following:

    • Multiply the expression by r

      • aₙ = a + ar + ar^2 + ar^3… + ar^(n-1) → raₙ = a + ar + ar^2 + ar^3… + ar^(n-1)

    • Subtract the second expression from the first

      • aₙ - raₙ = a - ar^n

    • Factor out aₙ from the left side

      • aₙ(1-r)=a-ar^n

    • Divide through by (1-r)

      • aₙ=(a(1-r^n))/(1-r)

  • If we want to find the sum of the first n terms of a geometric series, we use the following formula:

  • For an infinite geometric series, if it converges, lim (n→∞) r^n = 0, and its sum is found this way:

    • S = (a)/(1-r)

  • The nth Term Test For Divergence

  • The nth term test means we find the limit of the series and see if we get zero or not

    • If the limit = L (any number other than 0), the series diverges

    • If the limit = 0, we need to choose a different test

  • Integral Test for Convergence

    • If f is positive, continuous, and decreasing for x≥1, and aₙ=f(n), then

Either both converge or both diverge.

  • In other words, if you want to test convergence for a series with positive terms, evaluate the integral and see if it converges.

Harmonic Series & P-series

Harmonic Series: (1/n) pattern

  • Might look as though it is converging, but it diverges

  • The sums are not approaching a limit, so as the sequence of partial sus diverges, so does the sequence of partial sums for the harmonic series

p-Series

  • A series of the form (1/n^p) converges if p>1 and diverges if 0<p<1

Tests for Convergence

Comparison Tests for Convergence

  • Let 0 ≤ aₙ ≤ bₙ for all n

    • If the summation of bₙ from 1 to ∞ converges, then the summation of aₙ from 1 to ∞ converges

    • If the summation of aₙ from 1 to ∞ diverges, then the summation of bₙ from 1 to ∞ diverges

  • Essentially, if all of the terms of a series are less than those of a convergent series, then it too converges. If all of the terms of a series are greater than those of a divergent series, then it too diverges

Limit Comparison Test

Alternating Series Test for Convergence

  • Used for terms that alternate between positive and negative

  • The series using (-1)^(n+1) bₙ converges if the following conditions are all satisfied

    • bₙ > 0 (which means that the terms must alternate in sign)

    • bₙ > bₙ₊₁ for all n

    • bₙ → 0 as n → ∞

Absolute Convergence Theorem:

  • The series Σaₙ converges absolutely if the corresponding series Σ|aₙ| converges

Ratio Test for Convergence

Determining Absolute or Conditional Convergence

  • Sometimes, an alternating series will converge, but when we look at the absolute value of the terms, it doesn’t.

  • An alternating series that is not absolutely convergent is called conditionally convergent

Alternating Series Error Bound

  • When summing an alternating series with a finite number of terms, the sum will not be the same as the sum with an infinite number

  • If we want to find the error, we need to evaluate | S-Sₙ |

  • Can put a bound on the error: a number that the error is less than, which is simply the absolute value of the next term in the alternating series

    • If summing the first 10 terms in an alternating series, the error will be less than the absolute value of the eleventh term

Finding Taylor Polynomial Approximations of Functions

  • We can approximate a function on an interval centered at x=a by using a Taylor series (if centered about x=0, it’s a Maclurin series).

  • In a Taylor polynomial, the terms are found from thee derivatives of f as follows:

    • If f can be differentiated n times at a, then the nth Taylor polynomial about x=a is

    • pₙx = f(a) + f’(a)(x-a) + (f’’(a)/2!)(x-a)^2+ (f’’’(a)/3!)(x-a)^3+… + (f^(n)(a)/n!)(x-a)^n(x-a)^n

Lagrange Error Bound

  • The “error bound” of a Taylor polynomial, aka the Lagrange error

  • If you are finding an nth degree Taylor polynomial, a good approximation to the error bound is the next nonzero term in a decreasing series.

Radius and Interval of Convergence of Power Series

  • Going from n = 0 to n = ∞ instead of n = 1 to n = ∞

  • Radius of convergence: R.

    • If the series converges only for x=0, the radius of convergence is 0

    • If the series converges absolutely for all x, the radius is all x

    • Set of all values of X (-R, R) is the interval of convergence

Representing Functions as Power Series

  • We can use differentiation and integration of power series to find the power series of other functions

  • Can approximate the integral using its Maclurin and approximate the solution using the aforementioned series

  • Find a new Taylor series from a known Taylor series.

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