Question

Is the series geometric? If so, give the number of terms and the ratio between successive terms. If not, explain why not. $$ 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\cdots+\frac{1}{50} $$

Short answer

If yes, find the ratio and the number of terms. Answer: No, the given series is not a geometric series since the ratio between consecutive terms is not constant.

Step-by-step solution

Identify the series type

The given series is: $$ 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\cdots+\frac{1}{50} $$ We note that the series has a general term of the form: $$ a_n = \frac{1}{n} $$ with \(n\) ranging from \(1\) to \(50\).

Check if the series is geometric

In a geometric series, each consecutive term can be obtained by multiplying the previous term by a constant factor \(r\). If this series is geometric, we would have: $$ \frac{a_{n+1}}{a_n} = r $$ For all \(n\). Let's test if this holds: $$ \frac{a_{2}}{a_1} = \frac{\frac{1}{2}}{1} = \frac{1}{2} $$ and $$ \frac{a_{3}}{a_2} = \frac{\frac{1}{3}}{\frac{1}{2}} = \frac{2}{3} $$ Notice that the ratios are different — \(\frac{1}{2}\) and \(\frac{2}{3}\) —which means that the series is not geometric.

Conclusion

The given series is not a geometric series since the ratio between consecutive terms is not constant.

Key concepts

Series Analysis

The magic of mathematics often unravels through the study of series. A series is essentially the sum of a sequence of terms. In math, there are many types of series. Each possesses unique characteristics. This exercise involves the harmonic series, which is a specific kind of series. Understanding a series fully requires analyzing its nature.

**Harmonic Series**
A classic example of a series that isn't geometric is the harmonic series. Defined as the sum of reciprocals: \[ 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots \]it's properties differ from those found in other series. It's crucial to note that

• the harmonic series diverges, meaning it doesn't sum up to a finite number as more terms are added.
• the terms decrease in size but never reach zero.

**Analysis of Given Series**
Through analysis, if a series matches the harmonic pattern, we know it's not geometric. Given: \[ 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots + \frac{1}{50} \]Each term here is the reciprocal of an integer, solidifying its identity as a harmonic series.

Geometric Series

Unlike the harmonic series, a geometric series has a distinctive trait. It involves multiplying each term by a common factor. This factor is called the common ratio. When analyzing any series, seeing if it matches this pattern is critical.

**Characteristics of a Geometric Series**
A geometric series takes the form \[ a + ar + ar^2 + ar^3 + \ldots \]where:

• \(a\) is the first term of the series.
• \(r\) is the common ratio, identical for all terms.

For example, in a series like \[2, 6, 18, 54, \ldots \]each term is the previous term multiplied by 3, establishing it as geometric with a common ratio of 3.
To confirm a series as geometric, check the ratio \(\frac{a_{n+1}}{a_n}\).If it remains consistent, the series fits a geometric pattern. In the given series, this criterion fails, showing it's not geometric.

Common Ratio in Sequences

Determining if a series is geometric revolves around identifying a consistent multiplier. This constant factor is known as the common ratio.

**Understanding Common Ratio**
It's the backbone of geometric sequences. The common ratio \(r\) connects terms by the equation:\[\frac{a_{n+1}}{a_n} = r\]In a true geometric sequence, this formula yields the same number for all terms.

• If \(r = 1\), the series doesn't change.
• If \(r > 1\), the series grows.
• If \(0 < r < 1\), the terms shrink.

**Applying to the Given Series**
In our harmonic series example, the lack of a constant common ratio manifests:\[\frac{\frac{1}{2}}{1} eq \frac{\frac{1}{3}}{\frac{1}{2}}\]Consequently, it can't be labeled as a geometric series. Understanding how the common ratio operates here helps in clearly distinguishing between series types. This knowledge is key to mastering series analysis.