Taylor Series and Maclaurin Series - Calculus 2
The Organic Chemistry Tutor・2 minutes read
To find Maclaurin series, evaluate derivatives at C=1 and C=3 for Ln X and e^x, respectively, simplifying series for each function. The series for cosine X is derived from the series for sine X, involving alternating terms, derivatives, and factorial calculations.
Insights
- The Taylor series for Ln X centered at c=1 involves calculating derivatives of the function at c=1 and using the formula Σ[(x-1)^n/n!] from n=0 to ∞ with alternating signs (-1)^n to write out the series.
- The Maclaurin series for cosine X can be derived from the Maclaurin series for sine X by taking derivatives and manipulating the terms, resulting in a series represented by x to the 2n+1 times negative 1 to the n divided by 2n+1 factorial with alternating signs.
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Recent questions
How can I find a Taylor series?
By evaluating derivatives and simplifying the series.
What is the formula for a Taylor series?
F(x) = F(1) + F'(1)(x-1) + F''(1)(x-1)^2/2! + ...
How do I write a Maclaurin series?
Use the formula F(x) = F(0) + F'(0)x + F''(0)x^2/2! + ...
What is the Maclaurin series for cosine X?
Cosine X = 1 - x^2/2! + x^4/4! - x^6/6! + ...
How is the Maclaurin series for x cosine X derived?
Multiply the series for cosine X by x.
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Summary
00:00
Taylor and Maclaurin Series Simplified
- To find a Taylor series for Ln X centered at c=1, write out the first four derivatives: F'(x) = 1/X, F''(x) = -1/X^2, F'''(x) = 2/X^3, F''''(x) = -6/X^4.
- Evaluate the function and derivatives at C=1: F(1) = 0, F'(1) = 1, F''(1) = -1, F'''(1) = 2, F''''(1) = -6.
- Use the formula to write the Taylor series: F(x) = F(1) + F'(1)(x-1) + F''(1)(x-1)^2/2! + F'''(1)(x-1)^3/3! + ...
- Simplify the series for Ln X: F(x) = (x-1) - (x-1)^2/2 + (x-1)^3/3 - (x-1)^4/4 + ...
- Write the series in summation notation: F(x) = Σ[(x-1)^n/n!] from n=0 to ∞ with alternating signs (-1)^n.
- For e^x centered at C=3, find the derivatives: F'(x) = e^x, F''(x) = e^x, F'''(x) = e^x, F''''(x) = e^x.
- Evaluate the function and derivatives at C=3: F(3) = e^3, F'(3) = e^3, F''(3) = e^3, F'''(3) = e^3, F''''(3) = e^3.
- Write the Maclaurin series formula: F(x) = F(0) + F'(0)x + F''(0)x^2/2! + F'''(0)x^3/3! + ...
- Simplify the series for e^x: F(x) = x + x^2/2 + x^3/6 + x^4/24 + ...
- Represent the series in summation notation: F(x) = Σ[x^n/n!] from n=0 to ∞ with alternating signs (-1)^n.
- Derive the Maclaurin series for cosine X from the Maclaurin series for sine X: Cosine X = 1 - x^2/2! + x^4/4! - x^6/6! + ... using alternating signs (-1)^n.
20:23
Maclaurin Series for Trigonometric Functions
- The Maclaurin series for cosine X involves a series from 0 to infinity, represented by x to the 2n+1 times negative 1 to the n divided by 2n+1 factorial.
- By taking the derivative of both sides of the expression for sine X, the result simplifies to cosine X, with the derivative of x to the 2n+1 being 2n times x to the 2n.
- The Maclaurin series for cosine x squared is derived by replacing x with x squared in the original series, resulting in negative 1 to the n times x raised to the 4n divided by 2n factorial.
- To find the Maclaurin series for x cosine X, the original series for cosine X is multiplied by x, leading to the series of negative 1 to the n times x raised to the 2n+1 over 2n factorial.
