Expansion Using Maclaurin's Series Concept Part 2

TL;DR

Learn expansion formulas for exponential, sine, cosine, and tangent functions using Maclaurin series.

Transcript

hi everyone today we are going to discuss expansion using maclaurin series part second in part first we discuss expansion of e to the power x what is the formula for e to the power x so that is equal to this is 1 plus x plus x square upon 2 factorial plus x cube upon 3 factorial plus x raised to 4 upon 4 factorial plus and so on if we write nth ter... Read More

Key Insights

• 0️⃣ Maclaurin series is used to find the expansion of functions around zero.
• 🤘 Alternating signs are observed in the sine and cosine series.
• ☺️ Hyperbolic functions' expansions involve using the exponential functions e to the x and e to the -x.

Questions & Answers

Q: What is the formula for the expansion of e to the power x in Maclaurin series?

The formula is 1 + x + x^2/2! + x^3/3! + ... + x^n/n!.

Q: How are the formulas for sine and cosine series derived in Maclaurin series?

The sine and cosine series formulas are derived by replacing x with -x in the expansions, leading to alternating signs in the terms.

Q: What is the expansion of hyperbolic sine and cosine functions using Maclaurin series?

The hyperbolic sine and cosine expansions are derived by substituting the expansions of e to the x and e to the -x functions, respectively.

Q: How is the logarithmic series expansion achieved through Maclaurin series?

The logarithmic series expansion is derived by differentiating the log(1+x) function and substituting the respective derivatives of 0 in the Maclaurin series.

Summary & Key Takeaways

• Discusses the expansion of e to the power x using Maclaurin series.

• Explores the series expansions of sine, cosine, and tangent functions.

• Demonstrates expansion of hyperbolic sine, cosine, tangent functions, and logarithmic series.