Error Bound & Interval Of Series

Unit: Infinite Sequence & Series

Chapter: Error Bound & Interval of Series

Reference: – Taylor Series, Remainder term, Lagrange error Bound, Alternating series error bound, Integral test, Comparison test, Ratio test, Root test, Geometric series, Power series, Maclaurin series, Binomial series, Estimating series sums, Interval estimation techniques.

After studying this chapter, you should be able to:

• Introduction to Lagrange Error bound & Integral test.
• Geometric & Taylor series.
• Power, Maclaurin & Binomial series.
• Estimating series sums & Interval estimation techniques.

Introduction to Lagrange Error Bound & Integral Test

Lagrange Error Bound:

• The Lagrange error bound, also known as the remainder term, is used to estimate the error when approximating a function using a Taylor polynomial.
• It provides an upper bound on the absolute value of the difference between the actual value of the function and the value obtained by the Taylor polynomial.
• The Lagrange error bound is most commonly used when dealing with Taylor polynomials, where the error term depends on the (n+1)th derivative of the function.

Integral Test:

• The integral test is a method used to determine the convergence or divergence of a series by comparing it to the convergence or divergence of an integral.
• It states that if a series, ∑ a_n, is positive, continuous, and decreasing on the interval [1, ∞), then the series converges if and only if the corresponding improper integral, ∫ 1 to ∞ an (x) dx, converges.
• The integral test provides a useful tool for analyzing the convergence or divergence of a series whose terms are difficult to directly evaluate.
• If the integral of a function a(x) is divergent, then the corresponding series is also divergent.
• Similarly, if the integral of a function a(x) is convergent, it does not necessarily mean that the series is convergent. Additional conditions must be satisfied, such as the positivity and monotonicity of the terms.

Introduction to Geometric & Taylor series: –

Geometric Series:

1. A geometric series is a series in which each term is obtained by multiplying the previous term by a fixed, non-zero constant called the common ratio (r).
2. The general form of a geometric series is Σ(ar^n), where a is the first term and r is the common ratio.
3. A geometric series converges if the absolute value of the common ratio (|r|) is less than 1. In other words, -1 < r < 1.
4. The formula for the sum of a convergent geometric series is S = a / (1 – r), where S represents the sum.
5. If the common ratio is outside the range -1 < r < 1, the geometric series diverges.
6. The sum of an infinite geometric series is finite only when it converges; otherwise, it is said to be divergent.
7. If |r| = 1, the sum of the geometric series diverges except in the special case where r = 1, in which case the sum is infinite.
8. Geometric series often arise in various mathematical and real-life contexts, such as exponential growth and decay, compound interest, and population growth.

Taylor Series:

1. A Taylor series is a series in which most of the terms cancel each other out, resulting in a simplified sum.
2. The cancellation of terms in a telescoping series occurs when each term is expressed as a difference between two consecutive terms.
3. The Taylor effect arises from the pattern of cancellation, where most terms in the series eventually cancel out, leaving only a few terms.
4. Telescoping series are often characterized by a specific pattern in the terms, allowing for the cancellation to occur.
5. Telescoping series can be finite or infinite, depending on the behavior of the terms and the cancellation pattern.
6. When a telescoping series is finite, the sum can be found by simply evaluating the remaining terms after cancellation.
7. The convergence or divergence of a telescoping series can often be determined by examining the behavior of the terms as the number of terms increases.
8. The telescoping series provides a useful tool for evaluating sums and can be found in various areas of mathematics, such as calculus, algebraic manipulations, and engineering applications.

Power, Maclaurin & Binomial series

Power Series:

• A power series is an infinite series of the form Σ(cₙ(x-a)ⁿ), where cₙ represents the coefficients, x is variable, and a is the center of the series.
• Power series can represent a wide range of functions, including polynomial functions, exponential functions, trigonometric functions, and logarithmic functions.
• The convergence of a power series depends on its radius of convergence, which determines the interval within which the series converges.
• The radius of convergence can be determined using various tests such as the ratio test or the root test.
• If x lies within the interval of convergence, the power series represents the function it converges to.
• The sum of a power series is the function it represents within its interval of convergence.
• Power series can be differentiated and integrated term by term within their interval of convergence.
• Power series can be used to approximate functions by truncating the series at a certain term, which is useful for numerical calculations.

Maclaurin Series:

• A Maclaurin series is a specific type of power series where the center of the series is a = 0.
• It is named after the Scottish mathematician Colin Maclaurin.
• The Maclaurin series is particularly useful for approximating functions near the origin.
• The Maclaurin series for a function can be obtained by expanding the function as a power series, typically using Taylor's formula.
• The coefficients in the Maclaurin series can be calculated using the derivatives of the function evaluated at x = 0.

Binomial Series:

• A binomial series is a power series representation of a binomial expression.
• It is based on the binomial theorem, which states that (a + b)ⁿ can be expanded using binomial coefficients.
• The binomial series is of the form Σ( Cₙ(x-a)ⁿ ), where Cₙ represents the binomial coefficients.
• The binomial series is centered at a and converges for |x – a| < 1.
• The binomial series is often used to approximate functions involving binomial expressions, such as (1+x)ⁿ or (1-x)ⁿ.
• The binomial coefficients in the series can be calculated using combinations or Pascal's triangle.
• The number of terms in the binomial series corresponds to the exponent of the binomial expression.

Estimating Series Sums

• Estimating series sums involves finding an approximate value for the sum of an infinite series using a finite number of terms.
• The accuracy of the estimation depends on the number of terms included in the sum.
• The partial sum of a series is the sum of a certain number of terms in the series.
• Estimating series sums is particularly important when dealing with divergent series, where the sum does not approach a finite value.
• The error in estimating a series sum is the difference between the actual sum and the estimated sum using the partial sum.
• The error can be quantified using error bounds, such as the Lagrange error bound, which provides an upper limit on the error.
• The Lagrange error bound is often used with the Taylor series to estimate the error in approximating a function.
• The error in estimating a series sum can be reduced by including more terms in the partial sum. However, it is not always possible to obtain an exact sum for an infinite series.
• Different convergence tests, such as the ratio test or the integral test, can be used to determine whether a series converges or diverges, aiding in the estimation of its sum.
• Numerical methods, such as numerical integration or iterative algorithms, can be employed to estimate the sum of a series when an analytical solution is not readily available.

Taylor Series & Power Series

Taylor Series:

1. A Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.
2. The terms in the Taylor series are determined by evaluating the derivatives of the function at the center point a.
3. The Taylor series provides an approximation of a function around a specific point by using polynomials.
4. The accuracy of the approximation depends on the number of terms considered in the series. Including more terms improves the precision of the approximation.
5. Taylor series are often used to approximate functions, especially in situations where the function is difficult to evaluate directly or for numerical computations.
6. The Taylor series expansion of a function can be truncated to a finite number of terms to obtain an approximation of the function in a specific range.

Power Series:

1. A power series is a type of Taylor series where the center point is usually chosen to be x = 0.
2. The convergence of a power series depends on the values of x for which it converges.
3. The interval of convergence of a power series is the range of x-values for which the series converges.
4. The radius of convergence is a measure of how far away from the center point the series converges.
5. The radius of convergence can be determined using the Ratio Test or the Root Test.
6. Power series are often used in calculus to represent functions and to perform calculations such as differentiation and integration.

Example: – Consider the series Σ((-1)ⁿ / n²), where n starts from 1.

Solution: -Error Bound Estimation: To estimate the error bound when approximating the series sum using a partial sum, let's say we want to find the error when using the first 10 terms to approximate the sum.

The Lagrange error bound formula can be used in this case. Let's denote the nth term of the series by aₙ = ((-1)ⁿ / n²). The (n+1)th derivative of aₙ is given by aₙ⁽ⁿ⁺¹⁾ = ((-1)ⁿ / n²)⁽ⁿ⁺¹⁾.

Taking the (n+1)th derivative and evaluating it at a suitable value c between 1 and 10 (inclusive), we can find the maximum value of the (n+1)th derivative within that range.

Now, we need to find the maximum value of the (n+1)th derivative. Taking the derivative of aₙ⁽ⁿ⁺¹⁾ and simplifying, we have aₙ⁽ⁿ⁺¹⁾ = ((-1)ⁿ / n²)⁽ⁿ⁺¹⁾ = ((-1)ⁿ⁺¹ * (n+1)⁽ⁿ⁺¹⁾) / n²⁽ⁿ⁺¹⁾.

By analyzing the terms, we can see that as n increases, the absolute value of the terms decreases. Therefore, to obtain the maximum value, we take c = 10.

So, a₁₁⁽¹²⁾ = ((-1)¹²⁺¹ * (10+1)⁽¹²⁾) / 10²⁽¹²⁾ = (11⁽¹²⁾) / 10²⁽¹²⁾.

Using a calculator, we find that the value of a₁₁⁽¹²⁾ is approximately 0.0834.

Now, plugging this value into the Lagrange error bound formula, we have R₁₀(x) = (0.0834 * (x – 1)¹¹) / (11!).

Therefore, the error bound when using the first 10 terms of the series to approximate the sum is given by R₁₀(x).

Key Points

• The error bound is used to estimate the maximum possible error when approximating the sum of a series using a partial sum.
• It provides a measure of the accuracy of the approximation and helps quantify the difference between the actual sum and the estimated sum.
• The error bound is typically calculated using techniques like the Lagrange error bound or the alternating series error bound.
• The Lagrange error bound is commonly used with the Taylor series, while the alternating series error bound is employed with the alternating series.
• The error bound depends on the remaining terms in the series that are not included in the partial sum.
• A smaller error bound indicates a more accurate approximation of the series sum.
• The error bound can be reduced by including more terms in the partial sum, but it may not always be feasible to compute an exact sum for an infinite series.
• The error bound is often expressed as an inequality, such as |Rₙ(x)| ≤ E, where Rₙ(x) represents the remainder or error term and E is the error bound.
• The interval of convergence represents the range of values for which a power series converges to a valid function.
• It is determined by the values of x for which the series converges, diverges, or requires further investigation.
• The interval of convergence may include the center of the series or be centered around a specific value.
• The convergence of a power series can be analyzed using convergence tests such as the ratio test, root test, or integral test.
• The ratio test compares the absolute values of consecutive terms in the series to determine convergence or divergence.
• The root test examines the behavior of the nth root of the absolute values of the terms.
• The integral test compares the series to the convergence or divergence of a corresponding integral to establish the interval of convergence.