Sequences & Series PDF

Module 5: SEQUENCES & SERIES CE 2210 - Numerical Solutions to CE Problems Asst. Prof. Alex A. Santos Email:[email protected] Department of Civil Engineering Faculty of Engineering University of Santo Tomas 1st Term, AY2024-...

Module 5: SEQUENCES & SERIES CE 2210 - Numerical Solutions to CE Problems Asst. Prof. Alex A. Santos Email:[email protected] Department of Civil Engineering Faculty of Engineering University of Santo Tomas 1st Term, AY2024-2025 Learning Outcomes At the end of this session, students should be able to: 1. Review the basic concepts & applications of basic sequence & series (arithmetic, geometric & harmonic) 2. Understand more advanced concepts surrounding basic sequence & series. 3. Approximate a function using power series with emphasis on Taylor & Maclaurin series. 4. Approximate a periodic function using Fourier series. Sequences Sequences vs. Series Sigma Notation for Series Finite vs. Infinite Sequence Basic Sequences & Series Arithmetic Series Example: Note: If Sn has a limit as n → ∞, then the series converges, otherwise it diverges. Arithmetic Series Examples: Determine if the following sequences are arithmetic, and if so, determine the common difference, d. more examples here... Arithmetic Series Class Activity: 1. Find the 40th term of the sequence where a1 =4 and d = −2 2. Determine the formula for the sequence described in no. 1. 3. Find the sum of the first 20 multiples of 4. 4. Find the sum of the first 10 terms defined by an = 3n − 4 5. An auditorium has 30 rows of seats. There are 15 seats in the first row and then an additional seat added on each end in each row thereafter. What is the seating capacity of the auditorium? Geometric Series Geometric Series Geometric Series Sum of Converging Series: Geometric Series Example: more examples here... Harmonic Series ▶ Example: more examples here... Converging vs. Diverging Sequence Note: DNE → does not exist Converging Sequence Example: Diverging Sequence Example: Absolute vs. Conditional Series Convergence Diverging Series Example: Conditional Converging Series Example: Root Test: Converging & Diverging Series Root Test Example: more examples here... Series Convergence Tests Power Series Power Series Note: the interval of all x’s, including the endpoints if need be, for which the power series converges is called the interval of convergence of the series Power Series Illustration: Interval of Convergence Power Series Example: Exercise: Derive the power series for the function f (x) = x 2−4x2 Power Series Example: more examples here... Power Series: Taylor Series ▶ The basis of numerical differentiation is the Taylor series Taylor Series ▶ image shows sin(x) and its Taylor approximations by polynomials of various order ▶ As the degree of the Taylor polynomial rises, it approaches the correct function. Example: Taylor Series Problem: Use zero- through fourth -order Taylor series expansions to approximate the function f (x) = −0.1x4 − 0.15x3 − 0.5x2 − 0.25x + 1.2 from xi = 0 with h = 1. That is, predict the function’s value at xi+1 = 1. Class Activity 1. Given y ′′ = x + y − y 2 , y(0) = −1, y ′ (0) = 1. Derive the Taylor series polynomial. 2. The Maclaurin series expansion for cos x is 2 4 6 8 cos x = 1 − x2 + x4! − x6! + x8! −... Starting with the simplest version, cos x = 1, add terms one at a time to estimate cos( π 3 ). After each new term is added, compute the true and approximate percent relative errors. Add terms until the absolute value of the approximate error estimate is within an error tolerance of 0.0001. 3. evaluate the Taylor series for cos(x), sin(x), tan(x), and sin(x) − tan(x), in the vicinity of x = 0 4. Evaluate the Taylor series for ex and ln(x) in the vicinity of x = 1 5. For questions 2 & 3, how large is the error if h = 0.1 and we truncate the series at n=2 6. Use Taylor series expansions with n = 0 to 6 to approximate f (x) = cos x at xi+1 = π π 3 on the basis of the value of f (x) and its derivatives at xi = 4. Note that π π this means that h = 3 − 4 = 12π gm c 7. The velocity of a falling parachutist can be computed by v(t) = c (1 − e−( m )t ). Use a first-order error analysis to estimate the error of v at t = 6, if g = 9.8 and m = 50 but c = 12.5 ± 1.5. Periodic Function A periodic function also called a periodic waveform (or simply periodic wave), is a function that repeats its values at regular intervals or periods. Periodic Function Examples: Fourier Series - a representation of a function as a series of constants times sine and/or cosine functions of different frequencies Formula: Note: L = period of the function a0 = average value an & bn = Fourier coefficients Fourier Series Applications in Civil Engineering ▶ The most useful applications is solving vibration problems. Vibrations in mechanical systems or structures like bridges and buildings arise as periodic functions of time. ▶ Complex loading conditions (e.g., wind loads, seismic loads) can be represented using Fourier series, allowing for easier analysis of how structures respond to these loads over time. ▶ Fourier series are used to analyze seismic waves and their effects on structures. By understanding the frequency content of seismic waves, engineers can design structures to better withstand earthquakes. ▶ In the design of buildings and infrastructure, Fourier series can help model sound waves and their propagation, aiding in noise control and acoustic design. Fourier Series Example: Fourier Series Example (con’t:) Fourier Series Class Activity: 1. Use a continuous Fourier series to approximate the sawtooth wave below. Plot the first three terms along with the summation. 2. Use the continuous Fourier series to approximate the square or rectangular wave function given below: Odd & Even Functions Odd & Even Functions Example: Properties of An Even Function Properties of An Odd Function Relationship Bet. Even & Odd Functions to Fourier Series - We could simply obtain the Fourier series for a function, f (x) defined within the interval −a < x < a. In this case, c = −a and x + 2L = a, thus L = a, if we could identify whether f (x) is an even or odd function and use the properties of these functions in order to find the coefficients of Fourier series, a0 , an and bn References I Chapra, Steven (2021). Numerical Methods for Engineers (8th Ed. steemit (2017). Mathematics - Mathematical Analysis Series Convergence Tests. url: https: //steemit.com/mathematics/@drifter1/mathematics- mathematical-analysis-series-convergence-tests.