series
Cards (12)
- Method of differences:
- Rewrite the general term of the series as a difference of two or more terms to find the sum of finite series
- Write out the first and last three terms of the series to identify which terms cancel out and which do not
- Add together all terms that do not cancel out and simplify the result
- Higher derivatives:
- Finding higher derivatives involves differentiating a function as many times as required
- The nth derivative of y = f(x) is denoted as d^n y / dx^n or f^(n)(x)
- Maclaurin series:
- The Maclaurin series of a function is an infinite sum of terms that estimates the function around x = 0
- The more terms in the Maclaurin series, the better the approximation
- The Maclaurin series expansion for a function f(x) can be found using the formula: f(x) = f(0) + f'(0)x/1! + f''(0)x^2/2! + ... + f^(r)(0)x^r/r!
- The Maclaurin series is valid for values of x that cause the series to converge and when f(0), f'(0), f''(0), ... are finite
- Series expansions of compound functions:
- Standard Maclaurin expansions provided in the formula booklet can be used for compound functions
- These expansions are useful for finding series expansions of compound functions with tedious derivatives or involving products of functions
- The summand can be expressed using partial fractions as: 3(3šš + 1)(3šļæ½ļæ½ + 4) ā” š“ļæ½ļæ½/(3ļæ½ļ潚 + 1) + šµšµ/(3šš + 4)
- The terms of the series that don't cancel out are the first and last terms
- The series can be expressed as: ā“ 3(3šš + 1)(3šš + 4) = 1/4 - 1/(3ļæ½ļæ½ļæ½ļæ½ + 4)
- To find the series expansion of ln cos š„š„, differentiate the function four times:
- šš(š„š„) = ln cos š„š„
- ššā²(š„š„) = -sin š„š„ cos š„š„ = -tan š„š„
- ššā²ā²(š„š„) = -sec^2 š„š„
- ššā²ā²ā²(š„š„) = -2sec^2 š„š„ tan š„š„
- ššā²ā²ā²ā²(š„š„) = -2sec^4 ļæ½ļ潚„ - 4sec^2 ļæ½ļ潚„ tan^2 š„š„
- Plugging š„š„ = 0 into the derivatives:
- šļæ½ļæ½(0) = ln 1 = 0
- ššā²(0) = -tan(0) = 0
- ššā²ā²(0) = -sec^2(0) = -1
- ššā²ā²ā²(0) = -2sec^2(0) tan(0) = 0
- ššā²ā²ā²ā²(0) = -2sec^4(0) - 4sec^2(0) tan(0) = -2
- The series expansion of ln cos š„š„ is approximately -š„š„^2/2 - š„ļæ½ļæ½^4/12
- To find the expansion of ln(1 + š„š„), use the formula: ln(1 + š„š„) = š„š„ - š„ļæ½ļæ½^2/2 + š„š„^3/3
- To find the expansion of ln(1 - 3š„š„), use the formula: ln(1 - 3š„š„) = -3ļæ½ļ潚„ - 9š„š„^2/2 - 9š„š„^3
- The required expansion is approximately š„ļæ½ļæ½ - ļæ½ļ潚„^2 + 4š„š„^3/3
- Valid for -1/3 ā¤ š„ļæ½ļæ½ < 1/3