You provided a series:

$S = 1 - \frac{x^2}{2} + \frac{x^4}{4} - \frac{x^6}{6} + \cdots$

We are tasked with evaluating the sum of the first 10 terms of this series when $x = 4$.

The series seems to follow the pattern:

$S_n = 1 - \frac{x^2}{2} + \frac{x^4}{4} - \frac{x^6}{6} + \cdots + (-1)^{n+1} \frac{x^{2n}}{2n}$

Where $x = 4$ and $n = 10$.

Step-by-Step Evaluation:

Let’s compute the sum of the first 10 terms of the series for $x = 4$. The general term can be written as:

$T_k = (-1)^{k+1} \frac{x^{2k}}{2k}$

Substituting $x = 4$ into this expression gives:

$T_k = (-1)^{k+1} \frac{4^{2k}}{2k} = (-1)^{k+1} \frac{16^k}{2k}$

Now, calculate the sum of the first 10 terms for $x = 4$.

Let's compute it for each $k$ from 1 to 10:

• For $k = 1$: $T_1 = 1$
• For $k = 2$: $T_2 = -\frac{16}{2(2)} = -4$
• For $k = 3$: $T_3 = \frac{16^2}{2(3)} = \frac{256}{6} \approx 42.67$
• For $k = 4$: $T_4 = -\frac{16^3}{2(4)} = -\frac{4096}{8} = -512$
• For $k = 5$: $T_5 = \frac{16^4}{2(5)} = \frac{65536}{10} = 6553.6$
• For $k = 6$: $T_6 = -\frac{16^5}{2(6)} = -\frac{1048576}{12} \approx -87464.67$
• For $k = 7$: $T_7 = \frac{16^6}{2(7)} = \frac{16777216}{14} \approx 1198372.57$
• For $k = 8$: $T_8 = -\frac{16^7}{2(8)} = -\frac{268435456}{16} = -16777160$
• For $k = 9$: $T_9 = \frac{16^8}{2(9)} = \frac{4294967296}{18} \approx 238609849.78$
• For $k = 10$: $T_{10} = -\frac{16^9}{2(10)} = -\frac{68719476736}{20} = -3435973836.8$

Now sum all of these terms:

$S_{10} = 1 - 4 + 42.67 - 512 + 6553.6 - 87464.67 + 1198372.57 - 16777160 + 238609849.78 - 3435973836.8$

The sum is approximately:

$S_{10} \approx -3027428477.73$

Thus, the sum of the first 10 terms when $x = 4$ is roughly $-3,027,428,477.73$.

Would you like further details on how the series behaves, or do you have any questions about this computation?

Related Questions:

1. How does the value of the series change as $x$ increases?
2. What would happen if the number of terms was changed to $n = 5$?
3. How would you express this series for $x = 0$?
4. What is the convergence behavior of this series as $n$ becomes large?
5. How could this series be related to other mathematical functions, such as the Taylor series?

Tip: When working with alternating series, checking whether the series converges or diverges can be helpful in determining how many terms are necessary for a good approximation.