Series Convergence Test for ∑ 5^n/(4^n +3)

In summary, the series ##\sum 5^n/(4^n + 3)## does not converge because the limit of the terms does not equal zero. This is determined using L'hopital's rule, and it can also be shown through the comparison test and the geometric test.
  • #1
bigu01
57
0

Homework Statement


Which of the series, diverge or converge ∑ 5^n/(4^n +3 )


Homework Equations





The Attempt at a Solution

Taking the limit as n→∞ we have (5^n ln 5)/ (4^n ln 4) , my question is here how does it become like this, which part am I missing here?
 
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  • #2
Oh I see, they have used L'hopital rule since we got infinity over infinity
 
  • #3
bigu01 said:

Homework Statement


Which of the series, diverge or converge ∑ 5^n/(4^n +3 )


Homework Equations





The Attempt at a Solution

Taking the limit as n→∞ we have (5^n ln 5)/ (4^n ln 4) , my question is here how does it become like this, which part am I missing here?

A necessary condition for [itex]\sum a_n[/itex] to converge is [itex]\lim_{n \to \infty} a_n = 0[/itex] (it is not a sufficient condition; the series [itex]\sum n^{-1}[/itex] diverges). Here [tex]
\lim_{n \to \infty} \frac{5^n}{4^n + 3}
[/tex] is calculated using L'hopital's rule. Since the limit is not zero the sum does not converge.
 
  • #4
If ##\Sigma a_n## converges, then ##\lim(a_n) = 0##.

If ##\lim(a_n) ≠ 0##, then ##\Sigma a_n## diverges.

Alternatively, you could apply the comparison test + the geometric test with ##|r| ≥ 1##.
 

1. What is the difference between convergent and divergent?

Convergent and divergent refer to two types of evolution where species either become more similar or more different over time. Convergent evolution occurs when two unrelated species develop similar characteristics due to similar environmental pressures, while divergent evolution occurs when a single species diverges into two or more distinct species due to different environmental pressures.

2. How do you determine if a species is undergoing convergent or divergent evolution?

Scientists can determine if a species is undergoing convergent or divergent evolution by studying their physical characteristics, DNA sequences, and environmental factors. If two species have similar physical characteristics and share a common ancestor, it is likely an example of convergent evolution. If a single species diverges into two or more distinct species due to geographic or environmental factors, it is an example of divergent evolution.

3. Is there a clear distinction between convergent and divergent evolution?

While convergent and divergent evolution are two distinct types of evolution, there is often overlap between the two. In some cases, a species may exhibit both convergent and divergent characteristics. Additionally, the line between the two can be blurry, and it can be difficult to determine which type of evolution is occurring in a particular species.

4. What are some examples of convergent and divergent evolution?

An example of convergent evolution is the development of wings in birds, bats, and insects. While these species are unrelated, they have all evolved wings to adapt to their environment. An example of divergent evolution is the Galapagos finches, where a single species of finch diverged into multiple species with different beak shapes and sizes to adapt to different food sources on different islands.

5. Does convergent or divergent evolution have a greater impact on biodiversity?

Both convergent and divergent evolution play important roles in shaping biodiversity. Convergent evolution can lead to the development of new species with similar characteristics, while divergent evolution increases species diversity by creating new species with distinct characteristics. Both processes contribute to the overall biodiversity of our planet.

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