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For any real number p p, the series. is called a p-series. We know the p -series converges if p = 2 p = 2 and diverges if p =1 p = 1. What about other values of p? p? In general, it is difficult, if not impossible, to compute the exact value of most p p -series.
11 Ιουλ 2023 · In this chapter we introduce sequences and series. We discuss whether a sequence converges or diverges, is increasing or decreasing, or if the sequence is bounded. We will then define just what an infinite series is and discuss many of the basic concepts involved with series.
16 Νοε 2022 · Is the series a \(p\)-series (\(\sum {\frac{1}{{{n^p}}}} \)) or a geometric series (\(\sum\limits_{n = 0}^\infty {a{r^n}} \) or \(\sum\limits_{n = 1}^\infty {a{r^{n - 1}}} \))? If so use the fact that \(p\)-series will only converge if \(p > 1\) and a geometric series will only converge if \(\left| r \right| < 1\).
11 Ιουλ 2023 · Power Series and Functions – In this section we discuss how the formula for a convergent Geometric Series can be used to represent some functions as power series. To use the Geometric Series formula, the function must be able to be put into a specific form, which is often impossible.
Unit 10 - Infinite Sequences and Series (BC topics) 10.1 Defining Convergent and Divergent Infinite Series 10.2 Working with Geometric Series 10.3 The nth Term Test for Divergence 10.4 Integral Test for Convergence 10.5 Harmonic Series and p-Series 10.6 Comparison Tests for Convergence 10.7 Alternating Series Test for Convergence
Power Series – An introduction to power series and some of the basic concepts. Power Series and Functions – In this section we will start looking at how to find a power series representation of a function. Taylor Series – Here we will discuss how to find the Taylor/Maclaurin Series for a function.
A p-series is a series that involves n raised to a power p in the denominator of the nth term. This type of series is special because similar to a geometric series, we can easily determine its convergence.
