telescope

academic/misc/2024-math-calendar.html

@@ -191,7 +191,7 @@ <h1>Math Calendar 2024</h1>
         <td><a href="#mar26">26</a></td>
         <td><a href="#mar27">27</a></td>
         <td><a href="#mar28">28</a></td>
-        <td>29</td>
+        <td><a href="#mar29">29</a></td>
         <td>30</td>
         <td>31</td>
     </tr>
@@ -3548,6 +3548,21 @@ <h3 id="mar28">Mar 28</h3>
 100-50-33-14+16+7+4-2=100-97+27-2=28
 \end{align}\]
 
+<h3 id="mar29">Mar 29</h3>
+
+\[62\sum_{n=2}^{30}{1\over n(n+1)}\]
+
+<p>
+    Use partial fractions to rewrite \({1\over n(n+1)}={1\over n}-{1\over n+1}\)
+    and we see a telescoping series.
+</p>
+
+\[62\sum_{n=2}^{30}{1\over n(n+1)}
+=62\sum_{n=2}^{30}\left({1\over n}-{1\over n+1}\right)
+=62\left({1\over2}-{1\over3}+{1\over3}-{1\over4}
++\ldots+{1\over30}-{1\over31}\right)
+=62\left({1\over2}-{1\over31}\right)=31-2=29\]
+
 <!-- allow scroll past end a bit -->
 <div style="height:50vh;"></div>