Mathematics

Series Expansions and Coefficients

33 Questions

Series expansions involve representing functions as infinite sums, commonly using Taylor and Maclaurin series. This collection focuses on extracting coefficients and expanding binomials. These concepts are frequently tested in mathematics sections of various competitive exams.

Pascal's Triangle coefficientsMaclaurin seriesTaylor series expansionBinomial coefficients

Series Expansions and Coefficients Questions

Multiple choice

What is the coefficient of (x^4) in the expansion of ((3x^2-2)^{6})?

  1. 432

  2. 864

  3. 1296

  4. 1728
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Using the binomial theorem, we can expand ((3x^2-2)^{6}) as (\sum_{k=0}^{6} \binom{6}{k} (3x^2)^{6-k}(-2)^k). The coefficient of (x^4) corresponds to (k=2), so we have (\binom{6}{2} (3x^2)^{6-2}(-2)^2 = \frac{6!}{2!4!} \cdot 3^4x^8 \cdot 4 = 1296x^8).

Multiple choice

What is the exponential generating function for the Stirling numbers of the second kind?

  1. \(\frac{1}{(1-x)^n}\)
  2. \(\frac{1}{(1+x)^n}\)
  3. \(\frac{1}{(1-x^2)^n}\)
  4. \(\frac{1}{(1+x^2)^n}\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The exponential generating function for the Stirling numbers of the second kind is (\frac{1}{(1-x)^n}), which can be derived using combinatorial arguments or by solving the recurrence relation for (S(n, k)).

Multiple choice

Which row of Pascal's Triangle contains the coefficients of the expansion of (x + y)^6?

  1. 6th row

  2. 7th row

  3. 8th row

  4. 9th row
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The coefficients of the expansion of (x + y)^n are given by the entries in the (n + 1)th row of Pascal's Triangle. Therefore, the coefficients of the expansion of (x + y)^6 are given by the entries in the 7th row of Pascal's Triangle.