To test the convergence of the series $\sum_{n=1}^\infty u_n$, where

$u_n = \frac{1 + 3n}{1 + 2n},$

using partial sums, follow these steps:

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Step 1: Understand the Partial Sum

The $n$-th partial sum of the series is given by:

$S_n = \sum_{k=1}^n u_k.$

This represents the sum of the first $n$ terms of the series.

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Step 2: Express the General Term $u_n$

The term $u_n$ is already defined as:

$u_n = \frac{1 + 3n}{1 + 2n}.$

Simplify this term if possible to better analyze its behavior, but here it is already in reduced form.

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Step 3: Analyze the Behavior of $u_n$

Examine whether $u_n \to 0$ as $n \to \infty$, a necessary condition for the convergence of a series. Compute:

$\lim_{n \to \infty} u_n = \lim_{n \to \infty} \frac{1 + 3n}{1 + 2n}.$

Using the leading terms in the numerator and denominator, we find:

$\lim_{n \to \infty} u_n = \frac{3}{2}.$

Since $u_n \to \frac{3}{2} \neq 0$, the series diverges.

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Step 4: Confirm Using Partial Sums

Despite observing divergence based on the behavior of $u_n$, compute the partial sums $S_n$:

$S_n = \sum_{k=1}^n \frac{1 + 3k}{1 + 2k}.$

This involves summing a sequence of rational terms, but because $u_n$ does not approach zero, the partial sums $S_n$ grow unbounded.

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Step 5: Conclude the Convergence

Since $u_n \to \frac{3}{2} \neq 0$, the necessary condition for convergence is not satisfied. Therefore, the series $\sum_{n=1}^\infty u_n$ diverges.

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Do you want a step-by-step derivation of the partial sums $S_n$, or would you like to explore similar problems? 😊

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Follow-Up Questions

1. What does the divergence test state, and how does it relate to this problem?
2. How can one simplify expressions for general terms to analyze limits efficiently?
3. What is the difference between absolute and conditional convergence?
4. How do geometric and harmonic series compare to this example in terms of convergence?
5. What role does the growth rate of numerator and denominator play in determining series behavior?

Tip: Always check if the term $u_n \to 0$ as $n \to \infty$; it’s a quick test to rule out convergence!