8 Feb 2013

Prize Maths Quiz: Mercury and the Sum of a Series (PMQ5)

“Hello. Who are you?”

“My name is Alice. What’s yours?”

“I’m Mercury; welcome to my wedding.”

“Wedding?! You seem so young to be getting married!”

“I know, I’m supposed to look young, but actually I’m very old.” Mercury looked around as if he was about to slip out a secret. “All my family have been telling me for centuries to get married, settle down, raise some kids, but I just want to fly away.”

“Me too! I love to fly!!” Alice felt as if her words had just echoed around the whole universe and that everybody had heard them. She blushed a little.

At that moment, a thunderous clap jolted both Alice and Mercury from their reverie and they both turned their head to stare at what appeared to be the Sun slowly coming down to land in the middle of the... party! Alice had been so charmed by her new friend that she hadn’t noticed that she was in the middle of some kind of carnival.

As the figure descended from the heavens it became clear that this was no ordinary Sun, but a woman who shone like the brightest thing in the world. She was cloaked in a long gown the colour of the evening sky and embroidered with symbols and numbers.

“Who... is... that?” asked Alice in complete awe of the vision before her. Everyone bowed towards this special lady and Alice copied them.

“That is Arithmetica.” Explained Mercury; “She is the one, the two, the three and more. She is the infinite and the nothing. Even the gods bow down before her because without her they too would be nothing.” Mercury looked around, as if he was about to leave. “I think she has also brought her sisters along to remind us all about how important they are.”

As if in response to this whispered insult, Arithmetica was about to put on a show of power. From her forehead she emanated a dense beam of bright light. In mid-air the beam split into two as if forming two perpendicular axes. The light quickly split further in all directions until a web of light filled the entire wedding party. Then in the blink of an eye the whole lacework of laser beams contracted back into her forehead.

Alice thought her jaw was resting on the ground and tried to compose herself. For once in her life, she was quite literally speechless.

“Look, Alice, I hope you can stay for the wedding, but I really must go and take my official place. Let me guide you to a front seat.” Mercury took Alice’s hand and they glided towards the very bizarre-looking guests.

“Here, you can sit next to Pythagoras. You can talk to him about music.” And with that, Mercury flew away to take his place next to his wife-to-be, who immediately started nagging him. Poor Mercury! thought Alice.

The Question

Find the sum of the following infinite series:

You must include a valid method as well as the final sum.

How to Enter

Send your complete solution by email to [Competition now closed, so email removed. Feel free to discuss this question in the comments below.]. This email address shall be removed after the competition closes to avoid spam. This PMQ5 competition closes on Sunday 10 February at 23:59 GMT.

The Prize

The prizes for this PMQ5 are 3 free places in our Online Classroom for 2 months. The very first correct solution will receive a prize plus two others randomly selected from all the other correct answers. The email time stamp shall determine the order of entries received.
Quick Rules

Look at the expanded rules on our PMQ page.
Anybody can enter our Prize Maths Quiz; adults and students.
Use the official server clock in the right column to avoid late entries.
All emails and email addresses sent to us will be deleted after the winners have been processed.
DO NOT submit your entries to the comments section at the bottom of this post.
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